Seminar za teoriju vjerojatnosti

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Seminar za teoriju vjerojatnosti

Stranice Seminara za teoriju vjerojatnosti na PMF-u u Zagrebu.

Standardni termin održavanja seminara je utorak u 16 sati u učionici A002.

Voditelji seminara

Voditelji seminara: prof.dr.sc. Bojan Basrak, prof.dr.sc. Miljenko Huzak, prof.dr.sc. Hrvoje Šikić, prof.dr.sc. Zoran Vondraček

Tajnica seminara: Daniela Ivanković

seminari u prethodnim ak. god.

Seminare u prethodnim ak. god. možete pronaći ovdje.

sažeci za ak. god. 2024./2025.

O graničnom ponašanju geometrijskih funkcionala konveksnih ljuski slučajnih šetnji (1. dio)

U ovom prvom predavanju u nizu, proučavamo asimptotsko ponašanje konveksne ljuske koju generiraju međusobno nezavisne slučajne šetnje. Pokazujemo da, uz adekvatno skaliranje, ta konveksna ljuska gotovo sigurno konvergira prema konveksnoj ljusci generiranoj odgovarajućim vektorima drifta i ishodištem. Također dokazujemo gotovo sigurnu konvergenciju svih intrinzičnih volumena, što uključuje konvergenciju opsega i promjera. Nadalje, proučavamo distribucijsko ponašanje procesa opsega koristeći niz martingalnih razlika i Cauchyjevu formulu za kontrolu varijance. Na kraju dobivamo L^2 aproksimaciju za devijaciju opsega i utvrđujemo normalnu distribuciju procesa opsega pod određenim uvjetima.

Markovljevi procesi i Ramseyeva teorija za stabla

U ovom predavanju promatrat ćemo analogone Van der Waerdenovog (1927.) i Szemerédijevog (1975.) teorema gdje, umjesto aritmetičkih nizova, promatramo binarna stabla s fiksiranom udaljenošću između uzastopnih vrhova. Te verzije teorema pokazat će se jačima od prije spomenutih te ćemo za potrebe njihovih dokaza razviti inovativna svojstva povratnosti Markovljevih procesa. Predavanje je bazirano na članku Furstenberg, Weiss - Markov Processes and Ramsey Theory for Trees (2003).

sažeci za ak. god. 2023./2024.

Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box

In this talk, we will present a precise formulation of a conjecture regarding the product of the principal eigenvalue and the sup-norm of the landscape function in the discrete Anderson model restricted to a large box. First, we will show the asymptotic behavior of the principal eigenvalue as the box size increases. Then, we will use this result to prove the conjecture partially. In the one-dimensional case, we will provide a complete proof using Green function bounds. This talk is based on the work of Daniel Sánchez-Mendoza.

The Curse of Popularity: How High Degree Nodes in Graphs Distort the Relevance Estimated by Gaussian Random Projection

Random projections (RP) provide a simple and elegant approach to perform dimensionality reduction. In the context of large graphs, they can be applied to generate low-dimensional encodings of a node’s connectivity to other nodes in the graph. In this work, we provide an in-depth analysis of the behavior of three popular ways to measure relevance between the nodes estimated by Random Projections. We conclude that in two cases relevance estimation for high degree nodes will often be a significant miss. Such fail modes are important in practical applications, since high degree nodes often exert high influence in the graph. We develop results for the specific use of random projection in ranking and illustrate its consequences with a real graph dataset from Wikipedia. This work builds on many experiments the Graph Intelligence Sciences team at Microsoft did to compute relevance between entities (email, documents, people, events, ...) in Office 365. Joint work with Cassiano Becker and Jennifer Neville.

Poboljšane ograde za kose kutove

Kosi kut je konfiguracija različitih cjelobrojnih točaka oblika (x, y), (x, y+d), (x+d, y'). U ovom predavanju bavit ćemo se sljedećim pitanjem: koja je najveća veličina nekog podskupa kvadratne rešetke {1,...,n}^2 koji ne sadrži kosi kut? Ovo je pitanje nedavno postavio K. Pratt te je naslutio da vrijede vrlo jake gornje ograde na ovu veličinu. Štoviše, pomalo iznenađujuće, pokazao je da istinitost te slutnje povlači da se određena klasa brzih algoritama za množenje matrica ne može poboljšati. Međutim, njegova se slutnja ispostavila pogrešnom – nedavno su je opovrgnuli C. Pohoata i D. Zakharov. Mi ćemo pokazati konstrukciju koja značajno unaprjeđuje njihovu donju ogradu. Također, koristeći tehnike iz Fourierove analize vezane uz Rothov teorem o aritmetičkim nizovima, predstavit ćemo poboljšanje dosad poznatih gornjih ograda za ovaj problem. Komentirat ćemo i vezu s drugim srodnim problemima iz aditivne kombinatorike.

Paths in random temporal graphs

Random temporal graphs are a version of the classical Erdos-Rényi random graph G(n,p) where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time stamps. We are interested in the asymptotics for the distances in such graphs, mostly in the regime of interest where np is of order log n ('near' the phase transition). More specifically, we will discuss the asymptotic lengths of increasing paths: the lengths of the shortest and longest paths between typical vertices, as well as the maxima between any two vertices; this also covers the (temporal) diameter. In the regime np >> log n, longest increasing paths were studied by Angel, Ferber, Sudakov and Tassion. The talk contains joint work with Nicolas Broutin and Gábor Lugosi.

On the Limiting Behaviour of Geometric Functionals of Convex Hulls of Random Walk

We aim to deepen the understanding of geometric properties derived from independent random walks in the plane by establishing their pointwise and distributional limits. The primary goal is to demonstrate that individual random walks, when suitably scaled, almost surely converge to their drift vectors and, in a distributional sense, to a specified Brownian motion. This investigation will utilize the continuous mapping theorem to establish a broad spectrum of limit theorems for diverse geometric functionals. An illustrative focus of the research will be to show that the convex hull of multiple random walks almost surely converges to the convex hull of certain Brownian motions.

The study will also advance the analysis of distributional limits for the perimeter and diameter of convex hulls, drawing on the foundational work of Andrew Wade and his contemporaries. Techniques such as the Martingale Difference Sequence and Cauchy's formula will form the core methodological approach, aiming to model and understand the variance and behavior of geometric objects generated by random walks. The dissertation will explore the convex hull formed from the centers of mass points to demonstrate analogous results to those found for original convex hulls.

Teoremi gustoće za euklidske točkovne konfiguracije

Precizno ćemo formulirati rezultate koji garantiraju postojanje izvjesnih geometrijskih uzoraka u dovoljno velikim izmjerivim podskupovima euklidskog prostora. U drugom djelu predavanja ćemo se specijalizirati na situaciju u kojoj je promatrana konfiguracija 1-kostur jedinične hiperkocke te ćemo dati skicu dokaza teorema gustoće za tu konfiguraciju.

Primjene ergodske teorije u euklidskoj Ramseyjevoj teoriji

Cilj predavanja je proučiti kako pomoću ergodske teorije pristupiti određenoj klasi problema iz takozvane euklidske Ramseyjeve teorije. Ukoliko su nam zadani neki podskup euklidskog prostora te neka konfiguracija točaka, želimo odrediti pod kojim uvjetima možemo pronaći dovoljno velike izometrične kopije te konfiguracije unutar zadanog skupa. Pokazat ćemo kako se pitanja ovakvog geometrijskog tipa mogu pretvoriti u pitanja o rekurentnosti nekog skupa u pogodno izabranom sustavu koji čuva mjeru te kako se modificirani problem može riješiti pomoću tehnika ergodske teorije. Predavanje će se uglavnom bazirati na članku H. Furstenberg, Y. Katznelson, B. Weiss, Ergodic theory and configurations in sets of positive density, Mathematics of Ramsey theory, Algorithms Combin 5 (1990), 184–198.

Randomly augmented graphs

The model of randomly augmented graphs (also known as randomly perturbed graphs) combines two basic concepts studied in graph theory: Dirac type graphs and random graphs. Given an n-vertex graph G_α with minimum degree δ(G_α)>=αn, and a binomial random graph G_n,p, we call G_α U G_n,p a randomly augmented graph. This model has recently gained a lot of attention as a natural generalization of deterministic and random graphs. In my talk I will briefly survey the known results and methods used in studying randomly augmented graphs.

Sažeci za ak. god. 2022./2023.

Što sve (ne) znamo o bojenjima ravnine?

Razonodit ćemo slušatelja temom iz tzv. euklidske Ramseyeve teorije, koju su 70-tih godina prošlog stoljeća počeli sustavno proučavati Erdős, Graham, Montgomery, Rothschild, Spencer i Straus. Godine 1979. Erdős i Graham su pitali da li, za svako bojenje ravnine u konačno mnogo boja, nekoja boja sadrži vrhove pravokutnika bilo koje zadane površine. Taj je problem uvršten pod rednim brojem 189 na stranicu "Erdős problems": https://www.erdosproblems.com/189 Pokazat ćemo da je odgovor na ovo pitanje negativan, čak i za kvadre u više dimenzija, a jednostavnu konstrukciju izložit ćemo prema nedavnom preprintu: https://arxiv.org/abs/2309.09973 Potom ćemo spomenuti neke autorove povezane pozitivne rezultate koji koriste metode multilinearne Fourierove analize.

Dirichletove forme s jezgrom skokova degeneriranoj na granici

U ovom predavanju dat ću pregled nedavnih rezultata o Dirichletovim formama i pripadajućim Markovljevim procesima s jezgrom skokova degeneriranoj na granici. Objasnit ću opću teoriju kao i motivirajuće primjere, te opisati neke nove, neočekivane, značajke teorije potencijala i analize takvih Markovljevih procesa. Predavanje se temelji na nekoliko zajedničkih radova sa Soobin Choom, Panki Kimom i Renming Songom.

Podatkovni pristup u visokoškolskoj nastavi statistike

Bit će iznesene neke preporuke American Statistical Association-a kako držati visokoškolsku nastavu statistike imajući u vidu sve veću dostupnost velikih količina podataka.

Large Network Analysis using Random Projections

Information about entities in large networks is often represented as high dimensional numerical vectors. This makes calculating various relationships among entities computationally expensive. Ideally, we would like to compress the data we have and still be able to calculate the same relationships. Random projections have proven to be a powerful tool in dimension reduction and this approach simplified many problems . The Graph Intelligence Sciences team at Microsoft recently used this method to efficiently compute products of large matrices associated with Office 365 tenant-level graphs, and thus represent higher-order similarity between vertices in these graphs. In this talk, we will discuss how well some quantities are preserved under the random projection method and introduce new results on approximation quality under cosine similarity. We will also show some cases when the method can fail. Ongoing work with Cassiano Becker.

Regular variation on Polish spaces

We consider a general topological space X, endowed with an abstract family of bounded sets and a general notion of scaling. Within this framework, we define vague convergence and use it to define regularly varying measures and regularly varying random elements in X. In our analysis, we focus on the case where X is a Polish space. In particular, we investigate continuous mappings between Polish spaces that preserve the regular variation property. One notable mapping we explore is a novel variant of the polar decomposition that employs a modulus instead of a metric.

Poisson hulls

We introduce a hull operator on Poisson point processes, the easiest example being the convex hull of the support of a point process in Euclidean space. Assuming that the intensity measure of the process is known on the set generated by the hull operator, we discuss estimation of the expected linear statistics built on the Poisson process. In special cases, our general scheme yields an estimator of the volume of a convex body or an estimator of an integral of a H\"older function. We show that the estimation error is given by the Kabanov--Skorohod integral with respect to the underlying Poisson process. A crucial ingredient of our approach is a spatial Markov property of the underlying Poisson process with respect to the hull. We derive the rate of normal convergence for the estimation error, and illustrate it on an application to estimators of integrals of a H\"older function.

Teorija izleta Wright-Fisherovih difuzija

Wright-Fisherove difuzije su procesi koji modeliraju udio populacije koji sadrži promatrani gen i kao takve imaju prostor stanja $[0,1]$. Ponašanje takvih procesa na rubu intervala $[0,1]$ je od posebnog interesa, a ujedno je i komplicirano jer difuzijski koeficijent na rubu iznosi $0$ i nije Lipschitz neprekidan. Pristup koji predlažemo za bolje razumjevanje ponašanja Wright-Fisherovih difuzija je definiranje Poissonovog točkovnog procesa za izlete Wright-Fisherove difuzije koji započinju u rubnim točkama intervala $[0,1]$.

Positive self-similar Markov processes obtained by resurrection

In this talk I will consider positive self-similar Markov processes obtained by resurrecting a strictly α-stable process at their first exit time from (0,∞)(0,∞). These processes are constructed by using the Lamperti transform. I will explain their long term behavior and give conditions for absorption at 0 at finite time. In case the process is absorbed at 0, I will give a necessary and sufficient condition for the existence of a recurrent extension. The interest in such resurrected processes comes from the fact that their jump kernel may explode at 0. Joint work with Panki Kim (SNU) and Renming Song (Univ. Of Illinois).

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.

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.

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.

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.

sažeci za ak. god. 2021./2022.

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.

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.

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).

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.

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.

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).

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.

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