Abstracts

• Aleksandar Bulj Ocjene za homogene Fourierove multiplikatore i višeparametarsku maksimalnu Fourierovu restrikciju

  Fourierovi multiplikatori i teorija Fourierove restrikcije spadaju u centralna područja harmonijske analize i proučavaju se desetljećima. Ova disertacija doprinijet će boljem razumijevanju tih koncepata i riješiti neke otvorene probleme vezane uz njih. U prvom dijelu dokazat će asimptotski stroge ocjene za Fourierove multiplikatore s homogenim unimodularnim simbolima, odgovarajući pritom na dva otvorena pitanja u teoriji mutiplikatora postavljena od strane V. Mazye te O. Dragičevića, S. Petermichl i A. Volberga. U drugom dijelu dokazat će općeniti rezultat za multiparametarske maksimalne ocjene generalizirajući rezultat Christa i Kiseleva te posljedično dokazati multiparametarske maksimalne ocjene za Fourierovu restrikciju, koje prije ovog rada nisu bile poznate u više od dvije dimenzije.

• Andreja Vlahek Štrok Asimptotsko ponašanje aproksimativnog procjenitelja maksimalne vjerodostojnosti parametara pomaka u višedimenzionalnim eliptičkim difuzijama

  For fixed T, we analyze a $k$-dimensional vector stochastic differential equation over time interval $[0,T]$: $dX_t=\mu(X_t,\theta)dt+\nu(X_t)\sigma dW_t$, where $\mu(X_t,\theta)$ is a k-dimensional vector and $\nu(X_t)$ is a $k\times k$-dimensional matrix, both consisting of sufficiently smooth functions. Matrix of diffusion parameters $\sigma$ is known, and vector of drift parameters $\theta$ is unknown. We prove that approximate maximum likelihood estimator of drift parameter obtained from discrete observations $(X_{iΔn},0≤i≤n)$, when $Δn=T/n$ tends to zero, is locally asymptotic mixed normal with covariance matrix that depends on maximum likelihood estimator obtained from continuous observations $(X_t,0≤t≤T)$, and on path $(X_t,0≤t≤T)$. To prove the desired result, we emphasize the importance of the so-called uniform ellipticity condition of diffusion matrix.

• Tomislav Kralj Law of the iterated logarithm for a random Dirichlet series

  Let $\left(X_n\right)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with Rademacher distribution. Let $F(\sigma)=\sum_{n=1}^{\infty} X_n n^{-\sigma}$. The goal of this talk is to prove the LIL theorem for $F(\sigma)$ as $\sigma$ gets closer and closer to $1/2$ from the right, that means $$ \limsup _{\sigma \rightarrow 1 / 2^{+}} \frac{F(\sigma)}{\sqrt{2 \mathbb{E} F(\sigma)^2 \log \log \mathbb{E} F(\sigma)^2}}=1 $$ almost certainly. The talk relies on the paper of Marco Aymone, Susana Frometa and Ricardo Misturini from 2020.

• Darko Brborović Statistička analiza repnog ponašanja zavisnih nizova

  U ovom radu predstavljamo analizu repnog ponašanja slučajnih nizova dvodimenzionalnih vektora, što uključuje i slučaj stacionarnih $M$-zavisnih nizova, $M\in\mathbb{N}$. Često se asimptotska analiza repnog ponašanja slučajnih nizova obavlja u okviru teorije točkovnih procesa. Ipak, ukoliko je dostupan veći broj repnih događaja moguće je postići konvergenciju prema normalnoj distribuciji, kao što je to slučaj u ovom radu. Motivirani relativno novim rezultatima o permutacijskim testovima za korelaciju između dvije slučajne varijable, objavljenima u DiCiccio and Romano \cite{CR}, u ovom radu predstavljamo i analiziramo test repne zavisnosti za nezavisne i jednako distribuirane dvodimenzinalne slučajne vektore. Dodatno, predstavljamo i permutacijski test nezavisnosti za stacionarne $M$-zavisne slučajne nizove.

• Marina Dajaković Elements of renewal theory for cluster point processes

  The main goal of this thesis is to discuss renewal theorems in the framework of the cluster point processes on the real line. More precisely, under mild integrability assumptions, we describe the asymptotic distribution of the shifted renewal cluster point process and, as a consequence, we show that its mean measure on a bounded interval is asymptotically proportional to the length of the interval. Also, we give the equivalent version of the latter result that allows one to determine the asymptotic behaviour of the functions of special convolution form. These generalizations of the well-known results from the standard renewal theory are obtained using the methods and tools of modern probability theory, such as point process theory, especially vague convergence, and coupling method.

• Marina Dajaković Teoremi obnavljanja za točkovne procese s klasterima

  Pretpostavimo da oko svakog koraka T_i=X_0+X_1+…+X_i, i>=0 procesa obnavljanja, opažamo slučajan broj događaja rasutih oko T_i na način koji omogućava zavisnost o zadnjem vremenu međudolaska. Za takav proces, uz određene pretpostavke na prve momente, pokazujemo generalizirane verzije proširenog teorema obnavljanja, Blackwellovog teorema obnavljanja, elementarnog teorema obnavljanja i ključnog teorema obnavljanja. Primijenit ćemo dobivene rezultate na poseban Poissonov proces s klasterima koji je već proučavan u literaturi te ćemo predstaviti rezultate simulacijske studije.

• Drago Plečko From Statistical to Causal Inference in Fair Machine Learning and Intensive Care Medicine

  Large scale data collection is now prevalent in almost all aspects of society. Availability of such data, and the analyses performed on such data, allow us to discover new, previously unseen statistical patterns, or possibly remind ourselves of patterns that were already familiar before. In many cases, statistical associations observed in large scale data give rise to new scientific questions. For example, analyses of university records demonstrate a disparity in admission rates between male and female university applicants; analyses of criminal justice data show that racial minorities in the US are more likely to be jailed than the majority group. In an entirely different and unrelated context, that of intensive care unit (ICU) medicine, a large body of evidence shows that critically ill individuals with a high body mass index have a better chance of surviving their illness when admitted to the ICU, compared to their leaner counterparts (known as obesity paradox). Seemingly disconnected, the above mentioned phenomena raise a basic question in common: how did the disparity observed in the data come about in the first place? Can we connect the observed disparity to the causal mechanisms that are present in the real world, and that generate the observed disparity? Providing causal explanations of this kind is key for the scientific understanding of the observed disparities. In fact, the discussed phenomena are amenable to almost the same methodological toolkit, despite the fact that they arise from entirely different scientific domains. In this talk, we discuss the issues of gender or racial bias in datasets, from a causal perspective. These topics are studied under the rubric of fair machine learning, but could also be seen as epidemiology of discrimination. In addition to this, we study some epidemiological questions in ICU medicine, such as the obesity paradox, and describe accompanying computational and statistical methods that are useful in tackling ICU research questions.

• Ivan Biočić Semilinear equations for non-local operators

  Let $D\subset \R^d$, $d\ge2$, be an open bounded set, $f:D\times \R\to\R$ a function, and $L$ a non-local operator. In this thesis we deal with the following semilinear problem $Lu(x)=f(x,u(x))$, $x\in D$, where we also impose boundary conditions in $D^c$ and/or on $\partial D$, depending on the type of the non-local operator $L$. We are interested in two types of operators. The first type of the operator $L$ is an integro-differential operator which can be written in the form $L=\phi(-\Delta)$, where $\phi:(0,\infty)\to(0,\infty)$ is a complete Bernstein function satisfying the weak scaling condition at infinity. The operator $-phi(-\Delta)$ can be viewed as the infinitesimal generator of a subordinate Brownian motion where the subordinator has $\phi$ as its Laplace exponent. The second type of the operator $L$ can be written in the form $L=\phi(-\Delta_{|D})$ where, again, $\phi:(0,\infty)\to(0,\infty)$ is a complete Bernstein function satisfying the weak scaling condition at infinity. Here the operator can be viewed as the infinitesimal generator of a subordinate killed Brownian motion where the subordinator has $\phi$ as its Laplace exponent as we show in the thesis.

• Marina Dajaković Prošireni teorem obnavljanja za označene točkovne procese

  U ovom izlaganja promatrat ćemo označene točkovne procese kod kojih je moguća zavisnost između zadnjeg vremena međudolaska i oznake. Nadalje, korištenjem metoda i alata moderne teorije vjerojatnosti, kao što su teorija točkovnih procesa te metoda sparivanja, dokazat ćemo odgovarajuću verziju proširenog teorema obnavljanja.

• Peter Kevei Asymptotic behavior of the solution to the stochastic heat equation with Lévy noise

  We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a Lévy space-time white noise. For fixed time and space we determine the exact tail behavior of the solution both for light-tailed and for heavy-tailed Lévy jump measures. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. We also determine the almost-sure growth rate of the solution for any fixed time. This is joint work with Carsten Chong (Columbia).

• Ivan Biočić Semilinear Dirichlet problem for subordinate spectral Laplacian

  We study semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a boundary condition. The operators cover and extend the case of spectral fractional Laplacian. We also study harmonic functions related to the operator and boundary behaviour of Green and Poisson potentials.

• Petra Lazić Ergodičnost procesa difuzija

  Ova disertacija se bavi problemom ergodičnosti (stohastičke stabilnosti) široke klase procesa difuzija. Preciznije, to uključuje određivanje njihovih ekvilibrija (stacionarnih distribucija), ali i brzine kojom konvergiraju prema tim ekvilibrijima. Konvergencija se promatra s obzirom na neku određenu funkciju udaljenosti. U istraživanju naglasak je stavljen na kvantitativni aspekt ovog problema, odnosno na eksplicitne ocjene brzine konvergencije marginalnih distribucija prema invarijatnoj mjeri s obzirom na dvije funkcije udaljenosti: udaljenost totalne varijacije i klasu Wassersteinovih udaljenosti (koja predstavlja konvergenciju u nešto slabijem smislu). Kako je geometrijska ergodičnost dosta istražena u literaturi, cilj rada je pronaći oštre i opće uvjete koji garantiraju sub-geometrijsku ergodičnost te ih primijeniti na konkretnim primjerima. Prvi dio istraživanja se bavi klasičnim procesima difuzija, a drugi difuzijama sa slučajnim prebacivanjem (koji osim neprekidne, difuzijske komponente sadrže i drugu, diskretnu komponentu koja u slučajnim trenucima mijenja ponašanje procesa). U oba slučaja promatra se i klasa procesa sa skokovima.

• Ivana Slamić Maximal Cyclic Subspaces for Dual Integrable Representations

  In the theory of wavelets, the concept of maximal shift-invariant spaces plays an important role. Maximality is characterized by the strict positivity of the periodization function, the property which also appears in the characterization of several independence and basis related properties for the system of integer translates. In this talk, we consider the concept in a more general setting of LCA groups and unitary dual integrable representations. We describe the dual integrable triples which allow a decomposition of the Hilbert space into an orthogonal sum of n maximal cyclic subspaces and analyze how the questions concerned with maximality are reflected in redundancy and basis related properties of the generating orbit. Of particular interest is the case when n=1, i.e., when the generating orbit is complete in the whole space. The talk is based on a joint work with Hrvoje Šikić.

• Marina Dajaković Elementi teorije obnavljanja za točkovne procese s klasterima

  Teorija obnavljanja je važan dio moderne teorije vjerojatnosti sa širokom primjenom. Međutim, standardna teorija obnavljanja nije primjenjiva na događaje koji se pojavljuju u klasterima, što je čest slučaj u mnogim područjima primijenjene vjerojatnosti. Proučavat ćemo procese obnavljanja s klasterima na skupu realnih brojeva te planiramo, uz određene uvjete, proširiti klasične teoreme obnavljanja na procese ovog tipa (Blackwellov teorem obnavljanja i Ključni teorem obnavljanja), kao i opisati asimptotsku distribuciju pomaknutog točkovnog procesa s klasterima (Prošireni teorem obnavljanja).

• Petra Lazić Sub-geometric ergodicity of regime-switching diffusion processes

  In this talk, we discuss subgeometric ergodicity of a class of regime-switching diffusion processes. We derive conditions on the drift and diffusion coefficients which result in subgeometric ergodicity of the corresponding semigroup with respect to the total variation distance as well as a class of Wasserstein distances.

• Petra Lazić Ergodicity of diffusion processes

  In this talk, we discuss ergodicity properties of a diffusion process given through an It^o SDE. We identify conditions on the drift and diffusion coeffcients which result in sub-geometric ergodicity of the correspond- ing semigroup with respect to the total variation distance. We also prove sub-geometric contractivity and ergodicity of the semigroup under Wasserstein distances. Finally, we discuss sub-geometric ergodicity of two classes of Markov processes with jumps.

• Ana Martinčić Špoljarić Modelling of telomere loss process with applications

  Studying the mechanism of telomere shortening is important for understanding cell aging in biology. Mathematically, it is a process with a deterministic and stochastic component, where the main source of randomness comes from abrupt telomere shortening. The aim of the thesis is to generalise time - discrete processes of telomere length, with and without presence of telomerase, to a time - continuous processes in order to analyse their asymptotic properties. Depending on sources of randomness included in the model, processes with continuous trajectories or processes with jumps are met.

• Darko Brborović Permutacijski test asimptotske nezavisnosti

  Na seminaru će biti uveden pojam asimptotske nezavisnosti, a zatim predstavljen permutacijski test asimptotske nezavisnosti s pripadajućim rezultatima. Također će biti predstavljeni rezultati simulacijske studije predstavljenog testa, kao i jedan primjer primjene testa u financijama.

• Hrvoje Planinić Palm theory for extremes of time series

  We will discuss a framework for describing the asymptotic behavior of high-level exceedances of stationary time series whose finite-dimensional distributions are regularly varying and whose exceedances occur in clusters. The main tools are the theory of point processes and the notion of the so-called tail process. The latter allows one to fully describe the asymptotic distribution of the extremal clusters using the language of standard Palm theory. Finally, we will briefly discuss how this framework can be generalized to deal with extremes of random fields and even some problems in stochastic geometry.

• Soobin Cho General law of iterated logarithm for Markov processes

  In this talk, we discuss general criteria for liminf and limsup laws of iterated logarithm (LIL) for sample paths of continuous-time Markov processes. We establish LILs under two local assumptions near zero (or near infinity): two-sided estimates on the first exit time and uniform bounds on the tails of the jumping measure. Our results cover a large class of subordinated diffusion, jump processes with mixed polynomial growths, jump processes with singular jumping kernels and random conductance models with long range jumps. We also briefly discuss some LILs for additive functionals of those processes.

• Panki Kim Heat kernel upper bounds for symmetric Markov semigroups

  It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this talk, we discuss the equivalence among these and off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. We also discuss some applications too. This talk is based on joint works with Zhen-Qing Chen, Takashi Kumagai and Jian Wang.

• Nina Kamčev Common linear patterns are rare

  Several classical results in Ramsey theory (including famous theorems of Schur, van der Waerden, Rado) deal with finding monochromatic linear patterns in two-colourings of the integers. Our topic will be quantitative extensions of such results. A linear system L over F_q is common if the number of monochromatic solutions to L=0 in any two-colouring of F_qⁿ is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of F_qⁿ. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Fox, Pham and Zhao characterised common linear equations. I will talk about recent progress towards a classification of common systems of two or more linear equations. In particular, any system containing a four-term arithmetic progression is uncommon. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems. Joint work with Anita Liebenau and Natasha Morrison.

• Milosz Krupski Stochastic derivation of a Lévy driven fully nonlinear mean field game

  "Mean field games" are a type of a stochastic game, leading to a system of partial differential equations. They are supposed to describe the evolution of a system in which a large number of identical players take into account their joint distribution to make decisions (e.g. to model the behaviour of a crowd). The corresponding PDEs are a pair of a Hamilton-Jacobi-Bellman equation (decision-making) and a Fokker-Planck-Kolmogorov equation (evolution of the distribution). In a recent preprint (co-written with Indranil Chowdhury and Espen Jakobsen) we put forward a new model of mean field games, based on a control of a Lévy process. In this talk I will present a stochastic derivation of our model (the system of PDEs) and comment on how it corresponds to other formulations of mean field games.

• Tvrtko Tadić Fleming-Viot 2 particle system spine a Bessel process?

  It is known from previous results that a spine of a Fleming-Viot-type system of n particles whose distribution law is a discrete Markov chain, converges in distribution for large n to the same Markov chain conditioned never to hit 0. We study the same problem for continuous time Markov processes, in this case, Brownian motion. Recently, version of the law of the iterated logarithm was found for two particle spine. Since Brownian motion conditioned to never hit 0 is a Bessel(3) process for which LIL holds, we are investigating is the spine a Bessel process or not. The problem is that we only have a way of calculating the value of the spine process at a sequence of random times that is growing at an exponential rate. We will discuss some results that we obtained, as well as some failures in showing that the distribution in question is not that of a Bessel process. Joint work with Krzysztof Burdzy.

  Offline and online experimentation

  Companies with millions of users consume a lot of data and signals to build and make their product better. But how is this exactly done? We will present an offline and online experimentation approach that is used in today’s industry to produce better software, online services and digital ads.

• Bojan Basrak Limit Theorems for Branching Processes with Immigration in a Random Environment

  Based on the joint work with P.Kevei (Szeged). We explore mathematical properties of a class of branching processes which attracts considerable interest in applications. We further show how our results in this context provide a new insight into the classical theorem of Kesten, Kozlov and Spitzer concerning random walks in random environment.

• Bojan Basrak Limit Theorems for Branching Processes with Immigration in a Random Environment

  Based on the joint work with P.Kevei (Szeged). We explore mathematical properties of a class of branching processes which attracts considerable interest in applications. We further show how our results in this context provide a new insight into the classical theorem of Kesten, Kozlov and Spitzer concerning random walks in random environment.

• Zoran Vondraček Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary

  In this talk I will consider a Markov process in the halfspace with jumping kernel decaying at the boundary and critical killing potential. The goal is to explain the construction of the corrsponding Green function and to derive sharp two-sided estimates. These estimates imply anomalous boundary behavior for certain Green potentials. As an application of the Green function estimates, I will discuss validity of the boundary Harnack principle.

• Stjepan Šebek Limit theorems for the capacity of the range of stable random walks

  In this talk, we establish a central limit theorem and functional central limit theorem for the capacity of the range process for a class of d-dimensional symmetric alpha-stable random walks with the index satisfying d/alpha > 5/2.

• Indranil Chowdhury On Poincaré (Fractional) Inequalities on Unbounded Domains

  In this talk, I begin with the local Poincaré inequality, connected directly to the Dirichlet problem for Laplace operator. Then, I introduce the `fractional’ versions of this inequality. Regarding unbounded domains, first I prove a necessary condition concerning the inequalities to hold. I discuss the best constant problem for strip like domain and the connection with related eigenvalues. I will also present some sufficient conditions on domains for fractional Poincaré inequalities to hold.

• Ivana Katić Zakova transformacija

  Uvodimo operator $Z$ koji se naziva Zakova transformacija. Pokazat ćemo kako radimo s operatorom $Z$ (te s operatorom $\tilde{Z}$) te ćemo time pokazati da su Fourierova transformacija i njezin inverz unitarni kao izravna posljedica osnovnih svojstava Fourierovih redova.

• Mario Stipčić Multilinearni singularni integrali pridruženi hipergrafovima

  Ovaj rad doprinosi teoriji zapetljanih multilinearnih singularnih integralnih formi tako što daje prve karakterizacije $L^p$ omeđenosti dijadskih verzija tih formi pridruženih hipergrafovima. Nadalje, dokazuje prve težinske ocjene i dominaciju rijetkim operatorima za takve forme. Rad potom primjenjuje dobivene karakterizacije na jedan otvoreni problem u teoriji vjerojatnosti. Uvodi se pojam ergodičko-martingalnih paraprodukata i pokazuje njihovu omeđenost i konvergenciju u izvjesnom rasponu $L^p$ normi. To daje jedan mogući odgovor na klasično Kakutanijevo pitanje. Konačno, rad diskutira veze s lemama o uklanjanju iz aritmetičke kombinatorike i teorije grafova.

• Ivan Biočić Semilinearne jednadžbe za nelokalne operatore

  U disertaciji ćemo proučavati semilinearne jednadžbe za nelokalne operatore. Naglasak će biti na nelokalnim operatorima općenitijim od frakcionalnog Laplaceovog operatora, tj. naglasak će biti na integrodiferencijalnim operatorima koji generiraju subordinirano Brownovo gibanje i subordinirano ubijeno Brownovo gibanje. Iskazat ćemo dovoljne uvjete na rubne uvjete semilinearnog problema uz koje ćemo pokazati da problem ima rješenje, te ćemo pokazati u kojem je slučaju rješenje jedinstveno. Rješenje ćemo promatrati u distribucijskom i slabom dualnom smislu. Također, u slučaju kada promatramo semilinearan problem na omeđenoj $C^{1,1}$ domeni, dat ćemo i nužne uvjete za postojanje i jedinstvenost rješenja. Kako bismo uspjeli pokazati željene rezultate, upareno će se koristiti vjerojatnosne i analitičke metode iz teorije potencijala spomenutih procesa.

• Ivana Valentić Periodic homogenization for Levy-type processes (Periodička homogenizacija za procese Levyjevog tipa)

  Glavni cilj ove disertacije je diskutirati periodičku homogenizaciju pseudo-diferencijalnog operatora Levyjevog tipa. Naš pristup ovom problemu bazira se na vjerojatnosnim metodama. Preciznije, kao glavni rezultat dokazujemo da odgovarajuće centriran i skaliran proces Levyjevog tipa generiran takvim operatorom slabo konvergira prema Brownovom gibanju s kovarijacijskom matricom danom u terminima koeficijenata operatora. Posebno se koncentriramo na klasu procesa Levyjevog tipa koji dozvoljavaju samo “male skokove” i na klasu procesa difuzija s degeneriranim difuzijskim koeficijentom. Ti rezultati generaliziraju i produbljuju klasične i dobro poznate rezultate vezane uz periodičku homogenizaciju difuzije i procesa Levyjevog tipa u balansiranom obliku. Kako bismo razriješili ove probleme nužno je kombinirati vjerojatnosni i analitički pristup i metode, kao što su teorija semimartingala, teorija stohastičke stabilnosti i teorija integro-diferencijalnih jednadžbi.

• Mario Stipčić Povezanost leme o uklanjanju trokuta s teorijom hipergrafova

  Razmotrit ćemo povezanost leme o uklanjanju trokuta s fundamentalnim rezultatima iz teorije brojeva. Dodatno, proučit ćemo i analitičku varijantu leme te njenu generalizaciju iskazanu pomoću hipergrafa.

• Mario Stipčić T(1) teorem za zapetljane multilinearne singularne integralne forme i konvergencija ergodsko-martingalnih paraprodukata

  U prvom dijelu izlaganja razmatrat ćemo zapetljane multilinearne singularne integralne forme pridružene hipergrafovima u posebnom slučaju kada je integralna jezgra savršena dijadska Calderón-Zygmundova. Navest ćemo nekoliko tvrdnji koje su ekvivalentne L^p omeđenosti tih formi, a uključivat će uvjete T(1) tipa, vektore Muckhenhouptovih težina te rijetke multilinearne forme.
  U drugom dijelu izlaganja definirat ćemo ergodsko-martingalni paraprodukt te utvrditi raspon eksponenata za koji taj niz operatora konvergira. Time ćemo ponuditi pristup povezivanja ergodičkih usrednjenja i martingala, točnije teorema o konvergenciji potonjih dvaju nizova operatora.

• Mario Stipčić L^p omeđenost zapetljanih dijadskih singularnih integralnih formi pridruženih hipergrafovima

  Promatrat ćemo zapetljane singularne integralne forme koje su pridružene hipergrafovima. U posebnom slučaju, kada je integralna jezgra predstavljena produktom dijadskih Haarovih funkcija, pokazat ćemo L^p omeđenost u određenom rasponu eksponenata. U slučaju kada je zadana savršena dijadska Calderón-Zygmundova jezgra omeđenost u istom rasponu ekvivalentna je posebnim uvjetima, odnosno oblicima omeđenosti.

• Ivana Valentić CGT za procese Lévyjevog tipa s „malim skokovima"

  Glavni cilj ovog predavanja je diskutirati periodičku homogenizaciju pseudo-diferencijalnog operatora Lévyjevog tipa. Naš pristup ovom problemu bazirat će se na vjerojatnosnim metodama. Točnije, kao glavni rezultat dokazat ćemo da odgovarajuće centriran i skaliran proces Lévyjevog tipa generiran takvim operatorom slabo konvergira prema Brownovom gibanju s kovarijacijskom matricom danom u terminima koeficijenata operatora. Posebno ćemo se koncentrirati na klasu procesa Lévyjevog tipa koji dozvoljavaju samo „male skokove“. Ti rezultati će generalizirati i razraditi klasične i dobro poznate rezultate vezane uz periodičku homogenizaciju za procese difuzija i procese Lévyjevog tipa u balansiranom obliku.

• Ivana Valentić CGT za degenerirane difuzije s periodičnim koeficijentima i primjene u homogenizaciji linearnih PDJ-a

  Pokazat ćemo funkcionalni CGT za klasu degeneriranih difuzija s periodičnim koeficijentima i time generalizirati dobro poznate klasične rezultate za uniformno eliptične difuzije. Komentirat ćemo kako taj rezultat primjeniti na periodičku homogenizaciju eliptičke jednadžbe s rubnim uvjetom i paraboličke jednadžbe s početnim uvjetom.

• Mario Stipčić Multilinearni singularni integrali pridruženi hipergrafovima

  U ovom izlaganju promatrat ćemo rezultate iz teorije zapetljanih multilinearnih singularnih integralnih formi. Navest ćemo karakterizacije L^p omeđenosti dijadskih verzija tih formi pridruženih hipergrafovima koja će uključivati i prve težinske ocjene i dominaciju rijetkim operatorima za takve forme. Nadalje, uvest ćemo pojam ergodičko-martingalnih paraprodukata i pokazati njihovu omeđenost i konvergenciju u izvjesnom rasponu L^p normi. To će dati jedan mogući odgovor na klasično Kakutanijevo pitanje.

• Panki Kim Estimates on transition densities of subordinators with jumping density decaying in mixed polynomial orders

  In this talk, we discuss the sharp two-sided estimates on the transition densities for subordinators whose Lévy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on Lévy measures, we also discuss a precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more. This is a joint work with Soobin Cho.

• Ivana Valentić Periodička homogenizacija za procese Lévyjevog tipa

  Diskutirat ćemo periodičku homogenizaciju pseudo-diferencijalnog operatora Lévyjevog tipa. Naš pristup ovom problemu bazira se na vjerojatnosnim metodama. Točnije, kao glavni rezultat želimo dokazati da odgovarajuće centriran i skaliran proces Lévyjevog tipa generiran takvim operatorom slabo konvergira prema Brownovom gibanju s kovarijacijskom matricom danom u terminima koeficijenata operatora. Posebno ćemo se koncentrirati na klasu procesa Lévyjevog tipa koji dozvoljavaju samo „male skokove“ i na klasu procesa difuzija sa singularnim difuzijskim koeficijentom. Ti rezultati će generalizirati i razraditi klasične i dobro poznate rezultate vezane uz periodičku homogenizaciju za procese difuzija i procese Lévyjevog tipa u balansiranom obliku. Kako bismo razriješili ove probleme bit će nužno kombinirati vjerojatnosni i analitički pristup te metode kao što su teorija semimartingala, teorija stohastičke stabilnosti i teorija integro-diferencijalnih jednadžbi.

• Tvrtko Tadić Fixed Point Equation and Local Dependence Measure

  We study solutions to the stochastic fixed point equation X d= AX + B where the coefficients A and B are nonnegative random variables. We introduce the “local dependence measure” (LDM) and its Legendre-type transform to analyze the left tail behavior of the distribution of X. We discuss the relationship of LDM with earlier results on the stochastic fixed point equation and we apply LDM to prove a theorem on a Fleming-Viot-type process. Joint work with Krzysztof Burdzy and Bartosz Kołodziejek.

• Zoran Vondraček O teoriji potencijala Markovljevih procesa s jezgrom skokova koja opada na granici

  Promatramo $\beta$-stabilan proces u Euklidskom prostoru $\mathbb{R}^d$, $0 < \beta \le 2$, koji je ubijen nakon izlaska iz otvorenog skupa $D$. Ubijen proces je tada subordiniran pomoću nezavisnog $\gamma$-stabilnog subordinatora. Dobiven proces $Y^D$ zove se subordinirani ubijeni stabilni proces. U dva nedavna članka pokazano je da teorija potencijala tog procesa ima neka zanimljiva svojstva. Prvo takvo svojstvo je oblik jezgre skokova koja ovisi o udaljenosti točaka do granice. Drugo svojstvo je činjenica da za neke vrijednosti indeksa stabilnosti $\gamma$ ne vrijedi granični Harnackov princip. U prvom dijelu predavanja ću dati pregled tih rezultata. Drugi dio predavanja bit će posvećen tekućem radu na teoriji potencijala procesa skokova u otvorenom podskupu $D\subset \mathbb{R}^d$ definiranog pomoću jezgre skokova koja ne ovisi samo o udaljenosti među točkama, već i o udaljenosti tih točaka do grance skupa. Zajednički rad s Panki Kimom i Renming Songom.

• Petra Žugec Marked Poisson cluster processes and applications

  We study the asymptotic distribution of the total and maximal claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence arrival rate of the future claims. We find sufficient conditions under which the total and maximal claim amount converges in distribution. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes processes as our key example. Besides that, we try to clarify the notion of stochastic intensity which can be described in several different ways.

• Darko Brborović Statistička analiza repnog ponašanja zavisnih nizova

  Teoriju permutacijskih testova ćemo primijeniti na testiranje nezavisnosti dva slučajna niza s obzirom na njihovo repno ponašanje. Prezentirat ćemo slučaj nezavisnih i jednako distribuiranih varijabli te m-zavisnih slučajnih varijabli. Također, bit će predstavljeni rezultati simulacijske studije vezane uz ovaj problem.

• Viktoriya Knopova Simulations of Lévy processes

  We consider some theoretical methods of simulations of infinitely divisible random variables. Further, we discuss various methods of simulation of a Lévy process on a finite time interval. In particular, we discuss the series representation of processes, constructed from the related Poisson point process.

• Petra Žugec Asimptotsko ponašanje suma zahtjeva za isplatama u označenim Poissonovim procesima s klasterima

  Promatrat ćemo asimptotsko ponašanje suma zahtjeva za isplatama u označenom Poissonovom modelu s klasterima. Najprije ćemo formalno uvesti model. U tom modelu oznake određuju visinu i druge karakteristike individualnih zahtjeva i potencijalno mogu utjecati na intenzitet dolazaka zahtjeva u budućnosti. Prezentirat ćemo dovoljne uvjete pod kojima suma zahtjeva za isplatama zadovoljava centralni granični teorem ili alternativno konvergira po distribuciji stabilnoj slučajnoj varijabli. Primijenit ćemo prethodne rezultate na tri specijalna slučaja označenih Poissonovih modela s klasterima u kojima linearni označeni Hawkesovi procesi imaju ključnu ulogu. Na kraju ćemo spomenuti rezultate vezane uz asimptotsko ponašanje maksimalnih visina zahtjeva za isplatama.

• Petra Lazić Ergodičnost procesa difuzija

  Tema ove doktorske disertacije je problem ergodičnosti (stohastičke stabilnosti) procesa difuzija i procesa difuzija sa slučajnim prebacivanjem. Na početku predavanja dat ćemo motivaciju i objasniti pozadinu problema te dati pregled trenutno poznatih rezultata. Nakon toga ćemo objasniti probleme kojima ćemo se baviti u doktorskom radu, prikazati već dobivene rezultate te plan daljnjeg rada. U radu će naglasak biti na kvantitativnoj analizi problema ergodičnosti, odnosno na eksplicitnim ocjenama brzine konvergencije marginalnih raspodjela procesa prema pripadnoj stacionarnoj raspodjeli. Kombinirajući analitičke i vjerojatnosne tehnike, cilj rada je izvesti oštre i opće uvjete za subeksponencijalnu ergodičnost procesa s obzirom na udaljenost totalne varijacije i klasu Wassersteinovih udaljenosti u terminima koeficijenata samog procesa. Od posebnog značaja bit će klasa procesa sa singularnim koeficijentima, čime ćemo poopćiti neke već poznate rezultate. Također, gornje rezultate ćemo primijeniti na klasu procesa difuzija s malim skokovima te klasu Markovljevih procesa sa skokovima dobivenim koristeći tehniku subordiniranja u smislu Bochnera.

• Petra Žugec Stohastički intenzitet i Hawkesovi procesi

  Na početku izlaganja napravit ćemo uvod u Hawkesove procese te objasniti intuiciju vezanu uz stohastički intenzitet. Nakon kraćeg pregleda literature predstavit ćemo i reprezentaciju Hawkesovih procesa kao (označenih) Poissonovih procesa s klasterima. Zatim ćemo formalno uvesti teorijski okvir potreban za definiciju stohastičkog intenziteta označenog točkovnog procesa i pripadnog stohastičkog funkcionala. Iza toga ćemo objasniti definiciju Hawkesovog procesa pomoću stohastičkog intenziteta te razjasniti problem koji se pri tome javlja, a vezan je uz egzistenciju i jedinstvenost samih procesa. Na kraju ćemo prezentirati rezultat koji nam govori da Hawkesovi procesi zaista postoje (i jedinstveni su).

• Lavoslav Čaklović Mrežni pristup modeliranju izbora

  U pozadini većine metoda izbora stoji Aksiom Izbora (Luce) i produktna formula (napr. binomial i multinomijalni slučaj). Ovdje će biti govora o tome kad to nije slučaj. Jedan od načina je uspoređivanje slučajnih funkcija korisnosti, a drugi elementarniji slučaj pretpostavlja ulazni niz uspoređivanja u parovima koji ne generira konzistentan tok. Poopćenje tog pristupa je da se umjesto niza parova promatra niz uspoređenih parova, trojki, četvorki ... itd. Također će biti govora i o agregaciji podataka dobivenih iz više slučajnih funkcija korisnosti. Bit će demonstriran i softver temeljen na metodi potencijala prilagođen velikim rijetkim grafovima. Moguća primjena je: analiza kupnje proizvoda u marketingu, analiza anketnih upitnika u kojima se traži od ispitanika i stupanj sigurnosti za odabrani ponuđeni odgovor, upitnici tipa best-worst itd. Drugi tip agregacije odnosi se na distribuciju vrijednosti usporedbe u paru. To znači da za zadani par mogućih izbor postoji čitav niz vrijednosti (grupna odluka, streaming ...) koji generira multigraf s puno paralelnih lukova. To bi mogao biti slučaj prepoznavanja emocija na temelju spektralne analiza glasa. Problem je potpuno otvoren.

• Sidney Resnick Why Model the Growth of Networks?

  Social network modeling provides plenty of data, but realistic models for network growth must be simple if mathematical results are expected. We have used preferential attachment (PA) models with a small number of parameters in an attempt to strike a balance between the mathematics and the statistical fitting. The PA models struggle to match the data, but provide a context in which to test methods and analyze estimation techniques. Numerical summaries of network characteristics are often estimated using methods imported from classical statistics without real justification. For example, the Hill estimator coupled with a minimum distance threshold selection technique are commonly used. We discuss some attempts to justify and understand these estimation methods in the context of PA models. Without a model and its properties, there is no way to understand the limitation of estimation methods.

• Darko Brborović Randomizacijski test nezavisnosti zasnovan na repnom ponašanju slučajnih nizova

  Na početku seminara ćemo ukratko sažeti osnove randomizacijskih/permutacijskih testova. Osvrnut ćemo se na članak CiCiccio-Romano: Robust permutation tests for correlation and regression coefficients, a nakon toga ćemo formulirati i dokazati test nezavisnosti zasnovan na repnom ponašanju dva n.j.d procesa.

• Vjekoslav Kovač Varijacijske ocjene za paraprodukte martingala

  U uvodu ćemo izložiti klasične ocjene za martingale s diskretnim parametrom i neke njihove novije težinske varijante. Prisjetit ćemo se i varijacijskih ocjena Lepinglea i Bourgaina. Potom ćemo bilinearne varijacijske ocjene, koje su bili dokazali Muscalu, Tao i Thiele, poopćiti na općenite diskretne martingale obzirom na istu filtraciju. One će nam pak poslužiti za konstrukciju i ocjenu paraprodukta dvaju cadlag martingala s neprekidnim parametrom. Tako dobivenu finalnu ocjenu interpretirat ćemo u terminima teorije "martingalnih grubih putova" te prikazati kao van-dijagonalni rubni slučaj ocjena Friza i Victoira te Chevyreva i Friza.

• Vanja Wagner Nelokalne kvadratne forme s ograničenjem vidljivosti

  Tema ovog seminara su nelokalne kvadratne forme na podskupu $D$ Euklidskog prostora, koje dopuštaju skokove samo između parova točaka $(x, y) \in D \times D$ za koje je segment $[x,y]$ sadržan u skupu $D$. Diskutirat ćemo regularnost spomenutih Dirichletovih formi za specijalne klase mjera skokova i domena $D$, te time pokazati postojanje pripadnog čisto skokovitog procesa na $D$ s ograničenjem vidljivosti. Također, promatrat ćemo pripadnu Poincaréovu nejednakost u tzv. domenama u formi utega i pokazati da Poincaréova konstanta forme s ograničenjem vidljivosti ima difuzivan tip skaliranja.

• Hrvoje Šikić Bayesov serum istine: Igra dvoboja

• Hrvoje Šikić Bayesov serum istine i svojstvo poticanja na istinu

• Hrvoje Šikić Bayesov serum istine 1: Osnove

• Tvrtko Tadić Stochastic Fixed Point Equation: Positively Dependent Coefficients and Generalized IED-random variables

  In this talk we present the generalized family of Inverse Exponential Decay (IED) radnom variables and use them to obtain new results on the properties of the solution of the stochastic fixed point equation X =(d) AX+B. We also show that previous results for ARMA processes hold when the noise comes from this family. Using the Laplace transformation, we are able to obtain tail estimates for X in the case when A and B are nonnegative and positively quadrant dependent. Joni ongoing work with Krzysztof Burdzy.

  Geographical Area Estimation Using Gaussians and Points of Interest

  We present a method for estimating polygons of geographical areas from location points that are likely to be in that area. We view the importance of this method in the context of parts of inhabited places that are not well defined and the need to estimate the area for local search purposes. We will show a method based on gaussian heat kernel using points of interest from Microsoft Bing for several cities.

• Clayton Barnes Systems of diffusions interacting through their boundary local time

  In 2001, Frank Knight constructed a stochastic process modeling the one dimensional interaction of two particles, one being Newtonian (or inert) in the sense that it obeys Newton's laws of motion, and the other particle being Brownian. We construct a multi-particle analog using Skorohod maps. This multi-particle analog is a system of diffusions interacting through their local time. We use properties of Skorohod maps to characterize the hydrodynamic limit of the system as the number of Brownian particles approaches infinity. Our method gives existence and uniqueness for the resulting nonlinear PDE with free boundary condition.

• Wojciech Cygan Isotropic Markov semigroups on ultrametric spaces

  During two lectures (90 minutes each) we will present a smooth introduction to the theory of isotropic Markov semigroups on ultrametric spaces. An ultrametric space is a metrizable topological space such that the metric $d$ satisfies the so-called strong triangle inequality, that is \begin{align*} d(x,y) \leq \max \{ d(x,z), d(z,y) \}. \end{align*} This inequality has many important consequences and the most significant one is that two balls of the same radius are either disjoint or one is contained in the other. Thus, the collection of all balls with a fixed positive radius forms a countable partition of the space. A well-known example of an ultrametric space is the ring of $p$-adic numbers $\mathbb{Q}_p$. Such a topological structure allows us to construct an interesting class of Markov operators on ultrametric spaces. These operators were initially studied by Taibleson (1975) and Vladimirov (1988) and later by Bendikov, Grigor’yan, Pittet and Woess (2012, 2014). In the lectures we will first give a few examples of ultrametric spaces and next we will focus on a precise construction of Markov generators and corresponding semigroups of operators. Finally, we will indicate some directions for possible further investigations.

• Krzysztof Burdzy Twin peaks

  I will discuss some questions and results on random labelings of graphs conditioned on having a small number of peaks (local maxima). The main open question is to estimate the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks. Joint work with Soumik Pal.

• Panki Kim Heat kernel estimates for symmetric jump processes with general mixed polynomial growths

  In this talk, we discuss transition densities of pure jump symmetric Markov processes in $\mathbb{R}^{d}$, whose jumping kernels are comparable to radially symmetric functions with general mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at the infinity. This is a joint work with Joohak Bae, Jaehoon Kang and Jaehun Lee.

• Renming Song Well-posedness and long-time behavior of singular Langevin stochastic differential equations

  In this talk, I will present results from a recent joint paper with Longjie Xie on damped Langevin stochastic differential equations with singular velocity fields. The results include the strong well-posedness of such equations and the exponential ergodicity for the unique strong solution.

• John Einmahl Estimation of the marginal expected shortfall: the mean when a related variable is extreme

  Denote the loss return on the equity of a financial institution as X and that of the entire market as Y. For a given very small value of p>0, the marginal expected shortfall (MES) is defined as , where QY(1−p) is the (1−p)th quantile of the distribution of Y. The MES is an important factor when measuring the systemic risk of financial institutions. For a wide non‐parametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p↓0, as the sample size n→∞. Since we are in particular interested in the case p=O(1/n), we use extreme value techniques for deriving the estimator and its asymptotic behaviour. The finite sample performance of the estimator and the relevance of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large US investment banks.

• Holger Rootzén Multivariate peaks over thresholds modelling

• Chen Zhou The Bootstrap in Extreme Value Theory

  This paper develops a bootstrap analogue of the well-known asymptotic expansion of the tail quantile process in extreme value theory. One application of this result is to estimate the variance of estimators of the extreme value index such as the Probability Weighted Moment (PWM) estimator. The context is the block maxima method or the peak-over-threshold method.

• Viktoriya Knopova Analytic construction of Markov processes

  In this mini-course we consider the basic principles of the construction of the fundamental solution to a Cauchy problem for integro-differential operators. We discuss the basic steps of the construction: 1. Construction of the series representation of the solution 2. Verification 3. Uniqueness We discuss particular examples of the operators. Finally, we briefly discuss some versions of the method, and applications.

  Overview, Presentation slides

• Wojciech Cygan Heat content and its generalization for Lévy processes

  I will recall and discuss a notion of heat content related to Lévy processes in Euclidean spaces. To start with, I will present instructive examples including Brownian motion and stable processes, and next I will focus on the study of the small time behaviour of the heat content (and of its more general version) for a rich class of Lévy processes. The talk is based on the joint project with Tomasz Grzywny (Wrocław University of Science and Technology).

• Danijel Krizmanić Zajednička funkcionalna konvergencija procesa parcijalnih suma i maksimuma

  U ovom predavanju proučit ćemo zajedničku konvergenciju procesa parcijalnih suma i maksimuma za linearne procese s nezavisnim i jednako distribuiranim regularno varirajućim inovacijama. S određenim uvjetima na težine linearnog procesa dobit ćemo funkcionalni granični teorem obzirom na Skorohodovu slabu $M_{2}$ topologiju na prostoru $D([0,1], \mathbb{R}^{2})$.

• Tvrtko Tadić Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

  We study solutions to the stochastic fixed point equation X d=AX + B when the coefficients are nonnegative and B is an inverse exponential decay (IED) random variable. We provide theorems on the left tail of X which complement well- known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the IED class. We describe the lower envelope for these ARMA processes. Joint work with Krzysztof Burdzy.

• Petra Žugec Označeni Poissonovi procesi s klasterima i primjene
• Hrvoje Šikić BMO prostori i dekompozicija težina

  Dobro je poznata veza težina i BMO prostora. Garnett-Jonesov teorem pokazuje da je A(p)-svojstvo težine usko vezano s udaljenošću logaritma te težine od omeđenih funkcija u BMO prostoru. Pokazuje se da je ovaj teorem usko vezan s dekompozicijom A(2)-težina koju su otkrili Coifman i Rochberg. Pokazat ćemo da je i u apstraktnoj formi Garnett-Jonesov teorem ekvivalentan jednoj općenitijoj formi dekompozicije težina. Ovo je zajednički rad s Morten Nielsenom sa Sveučilišta u Aalborgu, Danska.

• Bojan Basrak O konvergenciji mjera i primjenama u vjerojatnosti
• Nikola Sandrić Ergodičnost generaliziranog Ornstein-Uhlenbeckovog procesa sa po dijelovima linearnim driftom

  U ovom predavanju diskutirat ćemo svojstva egodičnosti generaliziranog Ornstein-Uhlenbeckovog procesa sa po dijelovima linearnim driftom. Prikazat ćemo kvantitativne ocjene brzine konvergencije prema pripadnoj stacionarnoj distribuciji, obzirom na metriku totalne varijacije i Wassersteinovu metriku, u terminima parametara funkcije drifta, pripadne Levyeve mjere te kovarijacijske matrice.

• Ivan Papić Stohastički modeli u transformiranom vremenu: frakcionalne Pearsonove difuzije i odgođeni CAR procesi

  Stohastički modeli u transformiranom vremenu koji se razmatraju uključuju transformaciju slučajnog procesa u novi slučajni proces putem slučajnog vremena, dobivenog inverzom stabilnog subordinatora za koji se pretpostavlja da je nezavisan od početnog procesa. U prvom dijelu analizirat će se frakcionalne Pearsonove difuzije, tj. Pearsonove difuzije u transformiranom vremenu putem inverza standardnog stabilnog subordinatora. Eksplicitno će se izračunati spektralna reprezentacija prijelaznih funkcija gustoća frakcionalnih Pearsonovih difuzija s teškim repovima i jaka rješenja odgovarajućih vremenski - frakcionalnih Kolmogorovljevih jednadžbi unazad. U sljedećem koraku dokazat će se konvergencija specifično definiranih koreliranih slučajnih šetnji u neprekidnom vremenu prema frakcionalnim Pearsonovim difuzijama. U drugom dijelu razmatraju se autoregresivni procesi u neprekidnom vremenu koje je odgođeno inverzom subordinatora. Bit će predstavljena njihova korelacijska svojstva.

• Stjepan Šebek Subordinate random walks

  In this work we consider subordinate random walks constructed from an underlying simple random walk on the integer lattice by using discrete subordination defined through a Bernstein function. We assume that the Bernstein function satisfies weak scaling conditions at zero. Since subordinate random walks are in some way a discrete version of subordinate Brownian motions, we expect that some classical and very important results about subordinate Brownian motions have its discrete counterparts in terms of subordinate random walks. We will show that the Harnack inequality for subordinate random walks holds and we will find heat kernel estimates for subordinate random walks. We intend to generalize these results to subordinate random walks on graphs.

• Tvrtko Tadić Law of the iterated logarithm for the Fleming-Viot type process and the equation X=AX+B

  We study the movement of a two particle system on a positive part of the real line. If one of the particles hits 0 it dies, while the other particle splits into two. Both processes are driven by Brownian motion. We present the law of the iterated logarithm for this process. It turns out that the key to proving this result, is describing the behavior of the stochastic fixed point equation X=AX+B around zero when A and B are positive random variables.

  Tvrtko Tadić Introduction to simulations and statistical computing in the cloud

  Today, R is a standard open source used for doing simulations and statistical calculations for research and education. As the technology developed and data got bigger, the company Revolution Analytics started enriching R so that it can do parallel processing and process more data in a stable way. In 2015 Microsoft acquired Revolution Analytics and continued with this work. Many solutions have been built into other Microsoft software. For example, database SQL servers can now execute R code. In this presentation, we will introduce how one can simply run R code on the Microsoft Azure cloud and how this can be used in research and education (of probability and statistics).

• Jószef Lörinczi Long and short time properties of Lévy-type processes associated with non-local Schrödinger operators

  A non-local Schrödinger operator is the sum of a pseudo-differential operator and a multiplication operator called potential. The pseudo-differential operator is chosen in such a way that it generates a symmetric Lévy process, thus the Schrödinger operator corresponds to a perturbation of a Lévy process in a specific sense, giving rise to position-dependent drift and jump terms. Whenever the non-local Schrödinger operator has a ground state (eigenfunction at the bottom of its spectrum), the associated process has a stationary distribution. In this talk I will explain the construction of this jump process, and discuss its path regularity and long-time behaviour by using the properties of the ground state.

• Oliver Dragičević p-ellipticity
• Rudi Mrazović Aditivne trojke bijekcija

  Neka su $\pi_1$ i $\pi_2$ dvije nezavisno odabrane bijekcije $\{1, \dots, n\} \to \mathbb{Z}/n\mathbb{Z}$. Kolika je vjerojatnost da je $\pi_1+\pi_2$ također bijekcija? Ovaj je problem u literaturi bio promatran u različitim oblicima, te su za traženu vjerojatnost poznate brojne donje i gornje ograde. Prezentirat ćemo zajednički rad sa Seanom Eberhardom i Freddiejem Mannersom u kojem pronalazimo odgovor do na faktor $1 + o(1)$. Dokaz koristi Hardy-Littlewoodovu metodu kružnice iz analitičke teorije brojeva te neke metode iz aditivne kombinatorike.

• Vanja Wagner On purely discontinuous additive functionals of subordinate Brownian motions

  Neka je (A_t)_t≥0 potpuno diskontinuiran pozitivni aditivni funkcional za subordinirano Brownovo gibanje X=(X_t)_t≥0 oblika A_t= Σ_s≤t F(X_s-,X_s). Promatrat ćemo dovoljne uvjete na funkciju F za koje konačnost (g.s.) aditivnog funkcionala u beskonačnosti A_∞ povlači konačnost njegovog očekivanja. Nadalje, primijenit ćemo ovaj rezultat za proučavanje relativne entropije vjerojatnosne mjere P_x i vjerojatnosne mjere inducirane potpuno diskontinuiranom Girsanovljevom transformacijom procesa X. Dokazat ćemo spomenute rezultate pod pretpostavkom globalnog uvjeta skaliranja na Laplaceov eksponent pripadnog subordinatora. Seminar je baziran na zajedničkom članku sa Zoranom Vondračekom.

• Zoran Vondraček O graničnom ponašanju subordiniranih ubijenih Levyjevih procesa

  Neka je $Z$ subordinirano Brownonvo gibanje u $R^d$ pomoću subordinatora s Laplaceovim eksponentom $\phi$. Proces $Z$ ubijen je pri izlasku iz otvorenog skupa $D$, te je ubijen proces subordiniran pomoću subordinatora s Laplaceovim eksponentom $\psi$. Dobiven proces označen je s $Y^D$. Pretpostavljamo da i $\phi$ i $\psi$ zadovoljavaju slabe uvjete skaliranja u beskonačnosti. Cilj nam je proučiti teoriju potencijala procesa $Y^D$, specijalno granično ponašanje. Ako je $D$ gladak otvoren skup, izvest ćemo oštre ocjene Greenove funkcije za $Y^D$ i dokazati granični Harnackov princip s eksplicitnom stopom pada u blizini granice od $D$. Ti rezultati pokazuju da je granično ponašanje od $Y^D$ u potpunosti određeno s $\phi$. S druge strane, u unutrašnjosti od $D$, ponašanje od $Y^D$ određeno je kompozicijom $\psi \circ \phi$. Uz određene dodatne pretpostavke, pokazujemo i oštre ocjene za jezgru skokova od $Y^D$. U slučaju da je $D$ $\kappa$-debeli skup, pokazat ćemo da $Y^D$ zadovoljava Harnackovu nejednakost te granični Harnackov princip za funkcije harmonijske u otvorenom skupu $E$ strogo sadržanom u $D$.

• Nikola Sandrić A note on the Birkhoff ergodic theorem

  The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behavior of the time average of a (integrable with respect to the invariant measure) function along the trajectories of the process, starting from the invariant measure, is a.s. constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this seminar talk, under the assumptions that the underlying process is a Markov process on metric space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the L1-Wasserstein metric, we will show that the assertion of the Birkhoff ergodic theorem holds in probability and Lp, p \ge1, for any bounded Lipschitz function and any initial distribution of the process.

• Diogo Oliveira e Silva On the root uncertainty principle

  This talk will focus on a recent result of Bourgain, Clozel and Kahane. One of its versions states that a real-valued function which equals its Fourier transform and vanishes at the origin necessarily has a root which is larger than c>0, where the best constant c satisfies 0.41<c<0.64. A similar result holds in higher dimensions. I will show how to improve the one-dimensional result to 0.45<c<0.60, and the lower bound in higher dimensions. I will also argue that extremizers for this problem exist, and necessarily possess infinitely many double roots. Time permitting, I will make the connection to several related problems. This is joint work with Felipe Gonçalves and Stefan Steinerberger.

• Vjekoslav Kovač Democratic systems of translates on LCA groups

  We attempt to give characterizations of democratic property for systems of translates on a general locally compact abelian group, along a lattice in that group. That way we generalize results from the joint paper by Hernández, Nielsen, Šikić, and Soria on systems of integer translates. However, the main motivation for this work is to show that more operative characterizations exist for lattices with a particular algebraic structure. Certain basic notions from convex geometry, group theory, ergodic theory, and additive combinatorics arise naturally in our discussion. This is joint work with Hrvoje Šikić.

• Tvrtko Tadić Renewing Fleming-Viot-type particle system

  We consider a Fleming-Viot-type particle system consisting of independently moving particles that are killed on the boundary of a domain. At the time of death of a particle, another particle branches. We are considering a special case when the particles are driven by Brownian motion. We will show results on the long term behavior of the process. Ongoing joint work with Krzysztof Burdzy.

• Kristina Ana Škreb Tehnika Bellmanovih funkcija za multilinearne martingalne ocjene

  Srž tehnike Bellmanovih funkcija temelji se na radovima R. E. Bellmana iz stohastičke optimalne kontrole nastalih sredinom 20-og stoljeća. Ipak, tek D. L. Burkholder je primijenio tu tehniku na dokaze čisto teorijskih rezultata iz teorije vjerojatnosti i matematičke analize, tj. preciznije, na ocjene martingala. Nadalje, F. L. Nazarov, S. R. Treil i A. L. Volberg su tu tehniku prilagodili harmonijskoj analizi, tj. problemima ocjene normi integralnih operatora Razvojem moderne multilinearne analize pojavili su se problemi omeđenosti multilinearnih singularnih integralnih operatora za čiji dokaz se martingalni pristup čini najizgledniji. Stoga se prirodno postavilo pitanje daljnje prilagodbe tehnike Bellmanovih funkcija za multilinearne martingalne ocjene. Cilj ovog rada je razviti varijantu tehnike Bellmanovih funkcija potrebne za dokaz Lp ocjena za dva različita martingala, koji nisu nužno adaptirani obzirom na istu filtraciju. Dokazuju se nove martingalne ocjene koje se potom primjenjuju u različitim matematičim granama. Od primjena u teoriji vjerojatnosti, pokazuju se Lp ocjene za tzv. paraprodukt martingala s neprekidnim vremenom te daje konstrukcija stohastičkog integrala u jednom specifičnom kontekstu kada integrator nije nužno semimartingal. Od primjena u harmonijskoj analizi daje se alternativni dokaz omeđenosti paraprodukta obzirom na toplinski tok te se dobivaju nove ocjene za paraprodukte obzirom na dvije različite dilatacijske strukture.

• Hrvoje Planinić Point processes in the analysis of dependent data

  Point process theory is a useful tool in the analysis of extremal properties of stochastic processes. In particular, for i.i.d. sequences with regularly varying marginal distribution, point process convergence results are used to describe the asymptotic behaviour of partial sums and maxima of the sequence. Moreover, this approach can be used to analyze dependent data. In contrast to the independent case, when dependence is present, intuitively speaking, (really) large observations usually occur at nearby time instances, forming so-called clusters. Existing point process convergence results typically lose the information about the order at which large observations occur within the cluster. However, this order is extremely important when, for example, studying ruin probabilities or the asymptotic behaviour of record times. Our goal is to find a new type of point process convergence result which would preserve this kind of information. Furthermore, we aim to use this result to prove new functional limit theorems for partial sums and maxima of several important dependent time series models for which none of the exisiting limiting theory is applicable. Also, we will study possible applications of this point process result in the analysis of asymptotic properties of record times of a stationary sequence.

• Vjekoslav Kovač Vjerojatnosna varijanta Szemerédijeve leme o regularnosti i njezine primjene u kombinatorici 

  Započet ćemo s dva poznata teorema aditivne kombinatorike, Rothovim teoremom o aritmetičkim nizovima i teoremom o uglovima, koji će potom biti svedeni na rezultat iz teorije grafova. Daljnje poopćenje će biti iskazano vjerojatnosnim jezikom, kako bi ga se dokazalo korištenjem Taove varijante Szemerédijeve leme o regularnosti. Ta lema daje dekompoziciju proizvoljne slučajne varijable na "strukturirani dio", "pseudoslučajni dio" i "grešku", a prezentirat ćemo njezin potpuni dokaz. Seminar će biti iz literature, a bazirat će se na zajedničkom radu s Filipom Bosnićem (Universität Bielefeld).

• Ivan Papić Frakcionalne Pearsonove difuzije 

  Pearsonova difuzija je rješenje stohastičke diferencijalne jednadžbe s linearnim driftom i najviše kvadratnim koeficijentom difuzije, s Brownovim gibanjem kao pogonskim procesom. Frakcionalne Pearsonove difuzije definiraju se kao kompozicija Pearsonove difuzije i inverza stabilnog subordinatora koji modelira vrijeme. Prijelazne funkcije gustoće frakcionalne Pearsonove difuzije zadovoljavaju (frakcionalnu) Kolmogorovljevu jednadžbu unaprijed, odnosno unazad. Prezentirana su klasična rješenja pripadnih frakcionalnih Kolmogorovljevih jednadžbi. Nadalje, za razliku od Pearsonovih difuzija koje imaju "short-range" korelacijsku strukturu, frakcionalne Pearsonove difuzije imaju "long-range" korelacijsku strukturu koja opada kao opća potencija.

• Stjepan Šebek Harnackova nejednakost za subordinirane slučajne šetnje 

  U ovom predavanju ćemo poopćiti već poznati rezultat o Harnackovoj nejednakosti za harmonijske funkcije na slučaj subordiniranih slučajnih šetnji. Subordiniranjem slučajne šetnje dobivamo šetnju čiji korak može poprimiti beskonačno mnogo vrijednosti i nema konačnu varijancu. Za takve slučajne šetnje Harnackova nejednakost još nije pokazana.

• Hrvoje Planinić Granični teoremi za zavisne vremenske nizove teških repova 
• Kristina Ana Škreb Bellmanove funkcije za (dijadske) martingale 
• Martin Tautenhahn Wegner estimate for random Schrodinger operators with sign-indefinite and long-range correlated potentials
• Paolo Di Tella On the chaotic representation property of compensated-covariation stable families of martingales

  We study the chaotic representation property for certain families S of square integrable martingales on a finite time interval [0,T]. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family S of square integrable martingales having deterministic mutual predictable covariation. The main result is: If S is a compensated-covariation stable family of square integrable martingales such that is deterministic for all X,Y in S and, furthermore, the system of monomials generated by S is total in L^2(\Omega,F_T,P), then S possesses the chaotic representation property. We shall then give concrete examples in the case of Lévy processes. This talk is based on a joint paper with H.-J. Engelbert.

• Ivana Antoliš, Mario Stipčić, Dekompozicija operatora pomoću slučajnih dijadskih sistema

  Za neprekidne operatore standardno je promatrati dekompoziciju pomoću neprekidne ortonormirane baze, a u ovom seminaru prezentiramo alternativni pristup. Uvođenje vjerojatnosnog modela u obliku slučajnih pomaka dijadskih sistema za rastav Calderón–Zygmundovih operatora prvi je uveo Tuomas Hytönen. Objasnit ćemo glavne korake u tom dokazu te ponuditi usporedbu brzine konvergencije sa standardnim pristupom.

• Vanja Wagner, 3G nejednakost za subordinirano Brownovo gibanje i Harnackova nejednakost za pripadni cenzurirani proces

  U ovom predavanju promatrat ćemo subordinirano Brownovo gibanje $X$ čiji Laplaceov eksponent subordinatora zadovoljava određeno svojstvo skaliranja. Dokazat ćemo 3G nejednakost za Greenove funkcije procesa $X$ na $\kappa$-debelim ograničenim otvorenim skupovima. Korištenjem tog rezultata pokazat ćemo Harnackovu nejednakost za nenegativne harmonijske funkcije pripadnog cenzuriranog procesa $Y$ na nekom otvorenom skupu $D$.

• Vesna Gotovac, Testing equality in distribution of random convex compact sets via theory of random hedgehogs and $\mathfrak{N}$-distances
  Convex compact subsets of $\mathbb{R}^d$ have a nice property that they are determined by a type of real continuous functions on unit sphere in $\mathbb{R}^d$ called support functions. Space of convex compact subsets is equipped with Minkowski addition and multiplication by non negative real numbers and these operations correspond to operations on their support functions, but it doesn't form a vector space. Using the vector space of hedgehogs, which are defined as differences of convex compact sets, standard definition of random convex compact set can be expanded to random hedgehogs and their characteristic functions can be defined. Also, using the theory of $\mathfrak{N}$-distances, the newly defined terms can be used to derive a statistical test for testing equality in distribution of two random hedgehogs.
• Polona Durcik, Norm-variation of ergodic averages with respect to two commuting transformations
  We show a quantitative result on L^p-norm convergence of ergodic averages with respect to two commuting transformations, by counting their norm-jumps and bounding their norm-variation. We approach the problem via real harmonic analysis, using methods for bounding multilinear singular integrals with certain entangled structure. This is joint work with Vjekoslav Kovač, Kristina Ana Škreb and Christoph Thiele.
• Tonći Antunović, First passage percolation kao stohasticki model kompetitivnog sirenja
  U ovom predavanju cemo predstaviti dva stohasticka modela kompetitivnog sirenja izgradjena na klasicnom first passage percolation modelu. U prvom modelu cemo proucavati adaptaciju "Two type Richardson modela" (TTRM) Haggstroma i Pemantlea na konacne grafove i fokusirati se na slucaj uniformnog grafa na fiksnom broju vrhova. Prezentirat cemo vrlo precizne asimptotske rezultate o konacnom ishodu evolucije modela kada broj vrhova tezi u beskonacnost. U drugom dijelu cemo predstaviti ideju stacionariziranja modela iz statisticke fizike, i kao specijalan slucaj prezentirati poopcenje rezultata Deijfen i Haggstrom o modificiranom TTRM-u. Ukoliko vrijeme dopusti, spomenut cemo jos jedan model u kojem se stohasticko sirenje moze prirodno uloziti u evoluciju "Preferential Attachment" modela slucajnog grafa. Ovo su rezultati nekoliko projekata napravljenih u suradnji s koautorima Yael Dekel, Elchanan Mossel, Yuval Peres, Eviatar Procaccia i Miklos Racz.
• Tvrtko Tadić, Can one make a laser out of cardboard?
  We consider two dimensional and higher dimensional semi-infinite tubes made of "Lambertian" material, so that the distribution of the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. If the light source is far away from the opening of the tube then the exiting rays are (approximately) collimated in two dimensions but are not collimated in higher dimensions. Using the properties of the arccosine distribution we are able to analyze the tail of the movement in the direction of the higher dimensional tube and to obtain asymptotic results on the exiting properties for these high dimensions. Further, an observer looking into the higher dimensional tube will see "infinitely bright" spot at the center of vision. In other words, in three dimensions, the light brightness grows to infinity near the center as the light source moves away. On the way we will present some interesting general results on overshoot and undershoot of a random walk. Ongoing joint work with Krzysztof Burdzy.
• Cristina Benea, AKNS systems and Iterated Fourier Integrals
  Iterated Fourier Integrals provide the answer to a question regarding AKNS systems with data in L^2. The study of such systems for data in L^p with 1 \leq p
• Vesna Gotovac, Comparing random sets using rank envelope test on support functions of inner structure
  In the last years, modeling of random sets has became very popular. The models of random sets can describe and explain many events like behavior of cells in organisms, particles in materials, presence of different plants and many others. Even when these objects are three-dimensional, we often concentrate to two-dimensional modeling, because usually, we are either interested only in the projection of the objects to the plane (e.g. ground area of plants or trees) or we study only cross-sections of a mass which create planar formations, and suppose that the behavior of the studied object is stationary in the third dimension (e.g. organic cells or material particles). In this talk a new approach to testing if two realizations come from the same model of random sets will be presented. The approach is based on support functions of convex compact sets and envelope tests.
• Enkelejd Hashorva, Ruin Probability & Ruin Time Approximation for fBm Risk Model with Tax
  In this talk we are concerned with approximations of the ruin probability and the ruin times (first and last one) in the fractional Brownian motion risk model with tax. This is a special model of risk theory with interesting features for the theory of extremes of Gaussian processes. Our solution is based on a reformulation of the problem in terms of extremes of certain Gaussian random fields.
• Diana Rupčić, Laplaceova transformacija na konusima Banachovih prostora sa strukturom rešetke
  Fokus istraživanja je na karakterizaciji pozitivno definitnih funkcija pomoću Laplaceove transformacije mjere pri čemu je domena funkcije konveksan konus u Banachovom prostoru koji je ujedno i vektorska rešetka, ali ne nužno Banachova rešetka. Ispituju se uvjeti na početni prostor, konveksan konus te samu funkciju koji bi omogućili integralni prikaz pomoću Laplaceove transformacije mjere koja je definirana na dualu početnog prostora. Jedan od ciljeva je uspostaviti vezu između neprekidnosti pozitivno semidefinitne funkcije i koncentracije mjere na nekom manjem podskupu. Također se promatraju pozitivno semidefinitne funkcije koje imaju vrijednosti u prostoru operatora. U tom slučaju i reprezentirajuća mjera ima vrijednosti u konveksnom konusu koji pripada prostoru operatora i definira se kao prirodna generalizacija klasične pozitivne mjere s vrijednostima u konveksnom konusu nenegativnih brojeva.
• Ivan Veselić, Uniformni ergodički teoremi i Glivenko Cantellijev teorem
  Ergodički teoremi su poopćenja zakona velikih brojeva. S druge strane teorem Glivenka i Cantellija je zakon velikih brojeva za zakon empiričke raspodjele niza nezavisnih, identično raspodjeljenih slučajnih varijabli. Prezentirat ćemo neke situacije i rezultate koje spajaju ta dva aspekta teorije vjerojatnosti.
• Vanja Wagner, Censored Lévy and related processes
  
• Hrvoje Planinić, Rekordi i vremena rekorda
  
  Rekord niza slučajnih varijabli je vrijednost niza veća (ili manja) od svih prethodnih.  Slučajne varijable mogu predstavljati jačine potresa, potraživanja od osiguranja, cijene dionice, rezultate utrka i slično. Iz prethodnih primjera prirodno se nameću pitanja poput: Koliko često se događaju rekordi? Što možemo reći o samim iznosima rekorda?
  U seminaru ćemo proučiti ta, ali i neka druga pitanja u slučaju kada su slučajne varijable nezavisne i jednako distribuirane s neprekidnom funkcijom distribucije.
  Na kraju ćemo diskutirati o problemima na koje nailazimo kada su slučajne varijable zavisne.
• Panki Kim, Laws of the iterated logarithm for symmetric jump processes
  The law of the iterated logarithm describes the magnitude of the fluctuations of stochastic processes. The original statement of the law of the iterated logarithm for a random walk is due to A. Y. Khinchin (1924). In this talk, we discuss the laws of the iterated logarithm for a large class of symmetric jump processes. This is a joint work with Takashi Kumagai and Jian Wang.
  
• Stjepan Šebek, Jaki zakon za neke generalizirane procese urni
  Modeli urni je zajednički naziv za modele kod kojih imamo jednu ili više urni koje sadrže kuglice jedne ili više različitih boja. U diskretnim vremenskim trenucima u urnama radimo izmjene sadržaja u skladu s nekim unaprijed definiranim pravilima i pratimo kako se mijenja proporcija kuglica pojedinih boja u urni. Na taj način dobivamo diskretno-vremenski slučajni proces. Postoji mnogo različith modela urni u literaturi. U seminaru ćemo proučavati model kod kojeg imamo jednu urnu koja sadrži kuglice crvene i crne boje i kod kojeg nam je zadana neprekidna funkcija f : [0, 1] -> [0, 1] koja određuje pravila izmjene sadržaja urne na način da u n-tom koraku u urnu dodajemo crvenu kuglicu s vjerojatnošću f(X_n), a crnu s vjerojatnošću 1 - f(X_n), gdje je X_n proporcija crvenih kuglica u urni u trenutku n. Bavit ćemo se s dva bitna pitanja vezana uz taj proces. Kao prvo, uz koje uvjete na funkciju f naš proces (X_n) konvergira gotovo sigurno, a kao drugo, u slučaju kada se javlja konvergencija, koji je nosač varijable koja je gotovo sigurno limes našeg procesa?