2023-11-09 (czwartek), 12:15 - 14:00
Pavel Ievlev (Université de Lausanne)
Extremes of multivariate locally-additive Gaussian random fields
Streszczenie:
In this talk, I am going to present some of my recent results in joint work with Nikolai Kriukiv on the extremes of multivariate Gaussian random fields. I will begin with the 2019 paper by K. Dębicki, E. Hashorva, and L. Wang, which laid the groundwork for further investigations in the area of multivariate Gaussian extremes. I will explain that some of the assumptions of this paper may not hold in cases that are practically important, and I will discuss how these issues can be amended by considering second-order contributions — I will clarify this terminology during the talk. Next, we will explore what is, in a sense, the simplest extension of these results from processes (indexed by R) to fields (indexed by R^n), which we refer to as 'locally-additive'. As an application of this extension, I will present an exact asymptotic result for the probability that a real-valued process first hits a high positive barrier and then a low negative barrier within a finite time horizon.
2023-10-26 (czwartek), 12:15 - 14:00
prof. Krzysztof Dębicki (Uniwersytet Wrocławski)
Ekstrema pól gaussowskich z nie-addytywną strukturą zależności, z zastosowaniem do "performance tables" i wartości własnych gaussowskich macierzy unitarnych
Streszczenie:
Zbadamy dokładną asymptotykę rozkładu supremum pewnej klasy pól gaussowskich, które posiadają lokalnnie nie-addytywną strukturę zależności. Uzyskane wyniki zilustrujemy zastosowaniem do badania wartości własnych gaussowskich macierzy unitarnych. Wykład oparty jest na wspólnej pracy z Pengiem Liu (Univ. of Essex) i Longiem Bai (Univ of Nankai).
2023-06-29 (czwartek), 13:15 - 14:00
prof. Tomasz J. Kozubowski (University of Nevada, Reno)
The Classical Laplace Distribution: Fundamental Properties, Extensions, and Applications
Streszczenie:
We review basic facts about the classical Laplace distribution and its asymmetric generalization. Both distributions naturally arise in connection with random summation and quantile regression settings, and offer an attractive and flexible alternative to the normal (Gaussian) distribution in a variety of setting, where the assumptions of symmetry and short tail are too restrictive. The growing popularity of the Laplace-based models in recent years is due to their fundamental properties, which include a sharp peak at the mode, heavier than Gaussian tails, existence of all moments, infinite divisibility, and, most importantly, random stability and approximation of geometric sums. Since the latter arise quite naturally, these distributions provide useful models in diverse areas, such as biology, economics, engineering, finance, geosciences, and physics. We review fundamental properties of these models, which give insight into their applicability in these areas, and discuss their various extensions, including Laplace-based time series and stochastic processes. This is a joint work with K. Podgorski.
2023-06-29 (czwartek), 12:15 - 13:00
prof. Anna K. Panorska (University of Nevada, Reno)
The Greenwood statistic, stochastic dominance, clustering and heavy tails
Streszczenie:
The Greenwood statistic Tn and its functions, including sample coefficient of variation, often arise in testing exponentiality or detecting clustering or heterogeneity. We provide a general result describing stochastic behavior of Tn in response to stochastic behavior of the sample data. Our result provides a rigorous base for constructing tests and assuring that confidence regions are actually intervals for the tail parameter of many power-tail distributions. We also present a result explaining the connection between clustering and heaviness of tail for several classes of distributions and its extension to general heavy tailed families. Our results provide theoretical justification for Tn being an effective and commonly used statistic discriminating between regularity/uniformity and clustering in presence of heavy tails in applied sciences. We also note that the use of Greenwood statistic as a measure of heterogeneity or clustering is limited to data with large outliers, as opposed to those close to zero.
2023-06-15 (czwartek), 12:15 - 14:00
prof. Oleksandr Marynych (Taras Shevchenko National University of Kyiv)
A group theoretic approach to convex hulls and (K,G)-convex hulls: basic properties, examples and applications
Streszczenie:
The standard convex closed hull of a subset of the Euclidean space is defined as the intersection of all images, under the action of a group of rigid motions, of a half-space containing the given set. We propose a generalization of this classical notion, that we call a (K,H)-hull, and which is obtained from the above construction by replacing a half-space with some other convex closed subset K of the Euclidean space, and a group of rigid motions by a subset H of the group of invertible affine transformations. The above construction encompasses and generalizes several known models in convex stochastic geometry and allows us to gather them under a single umbrella.
2023-05-11 (czwartek), 12:15 - 14:00
prof. Tomasz Rolski (Uniwersytet Wrocławski)
Simulation of uniformly distributed points on some geometrical objects
Streszczenie:
We will survey various methods for simulation of uniformly distributed points on geometrical objects. In particular we consider d-dimensional balls, d-1-dimensional spheres. Interesting and not obvious problems appear when simulating random points on ellipses or ellipsoids. We conclude with a "numerical methods" for generating random points on parametrized objects like hyper-ellipsoids.
2023-03-30 (czwartek), 12:15 - 14:00
dr Marek Arendarczyk (Uniwersytet Wrocławski)
From slash distributions to generalized convolutions
Streszczenie:
An $\alpha$-slash distribution built upon a random variable $X$ is a heavy tailed distribution corresponding to $Y=X/U^{1/\alpha}$, where $U$ is standard uniform random variable, independent of $X$. We point out and explore a connection between $\alpha$-slash distributions, which are gaining popularity in statistical practice, and generalized convolutions, which come up in probability theory in connection with generalizations of the standard concept of convolution of probability measures. In particular, we show that the maximum of independent random variables with $\alpha$-slash distributions is also a random variable with an $\alpha$-slash distribution and discuss possible generalizations of this observation.
Our theoretical results are illustrated by several examples involving standard and novel probability distributions.
2023-03-02 (czwartek), 12:15 - 14:00
prof. Krzysztof Dębicki (Uniwersytet Wrocławski)
Ekstrema odbitych procesów gaussowskich w dyskretnym czasie
Streszczenie:
Dla odbitego (w 0) procesu gaussowskiego o stacjonarnych przyrostach, badamy asymptotyki ogona rozkładu supremum i infimum, gdy czas jest dyskretny. Referat oparty jest na wspólnej pracy z Grigorijem Jasnovidovem (University of Lausanne).
2022-12-01 (czwartek), 12:15 - 14:00
Alicja Kołodziejska (Uniwersytet Wrocławski)
Spacery losowe w rzadkim losowym środowisku
Streszczenie:
Spacer losowy w rzadkim losowym środowisku to prosty spacer na zbiorze liczb całkowitych. Środowisko, w którym się porusza, zadane jest przez losowy wybór punktów, w których prawdopodobieństwo pójścia w prawo jest również wybrane losowo; w pozostałych punktach spacer porusza się jak symetryczny spacer losowy. Podczas referatu przedstawię pokrótce dotychczasowe wyniki dotyczące tego modelu oraz opowiem o słabych twierdzeniach granicznych, jakie uzyskaliśmy w ramach wspólnej pracy z Dariuszem Buraczewskim i Piotrem Dyszewskim.
2022-11-17 (czwartek), 12:15 - 14:00
dr hab. Paweł Lorek (Uniwersytet Wrocławski)
Dualności w łańcuchach Markowa z częściowo uporządkowaną przestrzenią stanów
Streszczenie:
Wystąpienie będzie składało się z dwóch części.
Część 1. Będzie to przegląd wyników dotyczących:
i) dualności Siegmunda, która pozwala badać prawd. ruiny w modelach typu "zagadnienie ruiny gracza"; ii) tzw. strong stationary duality (SSD), która pozwala badać prędkość zbieżności do stacjonarności; iii) "Intertwining" -- pozwala na studiowanie czasu do pochłonięcia. Podamy idee konstrukcji takich łańcuchów dualnych w przypadku częściowo uporządkowanej przestrzeni stanów. Podamy np. wzory na prawdopodobieństwo wygrania w pewnych wielowymiarowych modelach, nowe wzory na wartość oczekiwaną czasu do pochłonięcia (oraz tzw. warunkowego czasu do pochłonięcia) w jednowymiarowych modelach. Wszystkie trzy dualności będą użyte do podania tzw. optymalnego czasu do stacjonarności dla symetrycznego błądzenie po okręgu -- co jest poprawieniem wyniku Diaconisa i Filla.
Część 2. Pokażemy jak można wykorzystać dualność Siegmunda i SSD do podania wzorów na prawdopodobieństwa przejścia między stanami w dokładnie $n$ krokach. Metodę zastosujemy do błądzenia po hiperkostce. Krótko opowiemy o jednym ze "skutków ubocznych", jakim jest pewna tożsamość algebraiczna, z której m.in. wynika nowy dowód na inną znaną tożsamość, a także nowa postać liczb Stirlinga drugiego rodzaju.
2022-10-27 (czwartek), 12:15 - 14:00
dr Michał Burdukiewicz (Autonomiczny Uniwersytet w Barcelonie/Uniwersytet Medyczny w Białymstoku)
Pułapki w analizie danych peptydowych
Streszczenie:
Peptydy to krótkie łańcuchy aminokwasowe, które pełnią wiele ważnych biologicznych funkcji (np. przeciwdrobnoustrojowe). Z tych względów badacze poszukują nowych peptydów o potencjalnie istotnych funkcjach, co jednak jest bardzo czaso- i kosztochłonne w warunkach laboratoryjnych. Dlatego też często stosuje się metody uczenia maszynowego, począwszy od modeli liniowych, po uczenie głębokie, które umożliwiają szybkie i tanie wykrywanie określonych grup peptydów w danych genomicznych i metagenomicznych. W mojej prezentacji pokazuję nie tylko proces tworzenia takich narzędzi, ale również problemy z tym powiązane takie jak brak odpowiednich danych (10.1093/nar/gkac882) oraz nierzetelne benchmarki (10.1093/bib/bbac343).
2022-10-20 (czwartek), 12:15 - 14:00
prof. Krzysztof Dębicki (Uniwersytet Wrocławski)
Prawdopodobieństwo symultanicznej ruiny w wielowymiarowym gaussowskim modelu ruiny
Streszczenie:
Zanalizujemy dokładną asymptotykę prawdopodobieństwa jednoczesnej ruiny
skorelowanych procesów ryzyka modelowanych przez procesy gaussowskie o stacjonarnych przyrostach
z liniowym dryfem. Przedstawione wyniki są owocem wspólnej pracy z Krzysztofem Bisewskim (Univ. of Lausanne)
oraz Nikolaiem Kriukovem (Univ. of Lausanne).
2022-10-13 (czwartek), 12:15 - 14:00
prof. Tomasz Rolski (Uniwersytet Wrocławski)
H-pochodne od sup-funkcjonałów od ułamkowego ruchu Browna
Streszczenie:
We consider a family of sup-functionals of (drifted) fractional Brownian motion with Hurst parameter
$H\in(0,1)$. This family includes, but is not limited to: expected value of the supremum
\begin{align*}
{\mathcal M}_H(T,a) := {\mathbf E}\Big(\sup_{t\in[0,T]} B_H(t) - at\Big),
\end{align*}
the Wills functional, and the Piterbarg-Pickands constant. Explicit formulas for the first derivative of these functionals as functions of Hurst parameter evaluated at $H=\tfrac{1}{2}$ are established.
2022-06-09 (czwartek), 12:15 - 14:00
Adam Kaszubowski (Uniwersytet Wrocławski)
Optimality of impulse control problem in refracted Lévy model with Parisian ruin and transaction costs
Streszczenie:
We will consider optimal dividend problem for a company, whose underlying process is modeled by the spectrally negative Lévy process. We will assume that after each payment there will be a transaction cost implied and as a result cumulative dividend process will be a pure jump process. Therefore, instead of the traditional barrier or refracted payment strategy, we will consider so-called impulse strategy. In addition, when the controlled risk process will be below zero there will be an additional drift imposed to save the process from falling into a Parisian-type ruin. This will imply that the controlled risk process will be driven by the refracted-type SDE.
Our aim will be to concentrate on proving the optimality of the impulse strategy with the use of standard steps of the Verification Lemma. We will show how and when natural assumptions arise and how they are connected with the scale functions.
The talk is based on the joint work with Irmina Czarna.
2022-06-02 (czwartek), 12:15 - 14:00
prof. Zbigniew J. Jurek (Uniwersytet Wrocławski)
Selfdecomposable variables, their background driving distributions(BDDF), log-gamma variables and some graphs
Streszczenie:
Selfdecomposable variables (distributions) or Lévy class L, arise as a natural generalization of the central limit theorem. It is a quite large class and includes many classical distributions such as stable, gamma, log-gamma, t-Student, logistic, stochastic area under planar Brownian motion, Bessel-K, Bessel densities, Fisher z-distribution, etc. All class L distributions admit the random integral representation - a random integral with respect to some Lévy process Y , called as background driving Lévy process, in short BDLP. Probability distribution of Y(1) is called background driving distribution, in short: BDDF. In the lecture we will present the formulas for BDDF (and for some variables) in a such way that might be more useful for a simulation.
References:
[1] zjj (2022) Theory Probab. Appl. vol. 67(1), pp. 105-117;
[2] zjj (2021) Mathematica Applicanda, vol. 49(2), pp. 85-109.
2022-05-26 (czwartek), 12:15 - 14:00
dr Klaudiusz Czudek (Uniwersytet Mikołaja Kopernika w Toruniu)
Central limit theorem for random circle rotations
Streszczenie:
I am going to talk about a Markov process arising when circle rotations are applied randomly. The central limit theorem for additive functionals of this random walk will be discussed. It depends on two factors: Diophantine properties of the angle of rotation and the regularity of an observable.
2022-05-19 (czwartek), 12:15 - 14:00
prof. Tomasz Rolski (Uniwersytet Wrocławski)
Pochdna funkcjonałów typu supremum dla ułamkowego ruchu Browna w H=1/2
Streszczenie:
We consider a family of sup-functionals of (drifted) fractional Brownian motion
with Hurst parameter H. This family includes, but is not limited to: expected value
of the supremum, expected workload, Wills functional, and Piterbarg-Pickands constant.
Explicit formulas for the derivatives of these functionals as functions of Hurst parameter
evaluated at H = 1/2 are established.
In order to derive these formulas, we develop the concept of derivatives of fractional alpha-stable
fields introduced by Stoev & Taqqu (2004) and
propose Paley-Wiener-Zygmund representation of fractional Brownian motion.
The talk is based on joint work with Krzysztof Bisewski and Krzysztof Debicki.
2022-05-05 (czwartek), 12:15 - 14:00
dr Piotr Dyszewski (Uniwersytet Wrocławski)
Branching random walks and point processes
Streszczenie:
We will investigate the convergence of a branching random walk from the viewpoint of random measures. We will show that the case of stretched exponential (Weibull) displacements provides a continuous transition between the domain of one big jump and its complement. The talk is based on a joint work with Nina Gantert (TUM).
2022-04-28 (czwartek), 12:15 - 14:00
prof. Krzysztof Dębicki (Uniwersytet Wrocławski)
Pochodna wartości oczekiwanej supremum ułamkowego ruchu Browna w H=1
Streszczenie:
Supremum S(T,H) ułamkowego ruchu Browna z parametrem Hursta H na odcinku [0,T] gra istotną rolę w wielu modelach probabilistycznych. Rozkład zmiennej losowej S(T,H) znany jest jedynie gdy H=1/2 lub H=1. W wykładzie skupimy się na własnościach pochodnej względem parametru H wartości oczekiwanej S(T,H) w H=1. Przedstawione wyniki oparte są na wspólnej pracy z Krzysztofem Bisewskim (University of Lausanne) i Tomaszem Rolskim.
2022-04-21 (czwartek), 12:15 - 14:00
dr Konrad Krystecki (Uniwersytet Wrocławski)
Componentwise Finite-Time Ruin In Two-Dimensional Brownian Risk Model
Streszczenie:
The aim of this dissertation is to study a particular finite-time risk model for the ruin probability of
two insurance companies that partially share the same risk. The model assumes that the company’s cash flows are modelled by a random and a deterministic part. The accumulated claims are modelled by the Brownian motion and the deterministic premiums are modelled by the linear drift. The connection between the two companies is modeled through a constant correlation. In the first part the company is considered to be ruined if the cash flows of the company at any point in the given interval overcome the initial capital of the company. The thesis investigates the probability of both companies being ruined as their initial capitals are growing larger. In the subsequent parts we consider generalizations of the aforementioned notion of ruin. The new definition of ruin states that the ruin occurs only if the cash flows of the company are above the initial capital for a certain period of time, which is called time in red. We differentiate two cases – where we either count the time in red consecutively or cumulatively. The dissertation focuses on a particular length of the time that the cash flows spend above the initial capital, which is connected to the height of initial capital through a simple function. We prove that for that particular function we obtain results that are of the same order as the results from the first part. In the final chapter the thesis considers much more general cases of the time in red for the case of positively correlated companies.
2022-01-20 (czwartek), 12:15 - 14:00
dr Witold Świątkowski (Uniwersytet Wrocławski)
Norms of randomized circulant matrices
Streszczenie:
Opowiem o oszacowaniach norm macierzy losowych z niezależnymi współrzędnymi. Zacznę od zaprezentowania twierdzenia A. Bandeiry, R. Latały i R. van Handela, dającego obustronne oszacowanie normy, niezależne od wymiaru, w przypadku gaussowskim. Następnie zajmiemy się przypadkiem zero-jedynkowym. Pokażę, że zachodzi podobne oszacowanie dolne. Opowiem też, co już wiemy, a co chcielibyśmy wiedzieć o górnym oszacowaniu.
2022-01-13 (czwartek), 12:15 - 14:00
prof. Alexander Iksanov (Taras Shevchenko National University of Kyiv)
On the number of occupied boxes in an infinite occupancy scheme
Streszczenie:
I intend to provide an elementary introduction into an infinite
balls-in-boxes scheme a.k.a. Karlin's occupancy scheme. Also, two
generalizations of the scheme will be discussed. The core of the talk
will be devoted to weak convergence of the number of occupied boxes.
After presenting Dutko's (1989) criterion for the one-dimensional
convergence I shall turn to more recent advances concerning weak
convergence of finite-dimensional distributions. The talk should be
accessible to a wide audience with some knowledge of basic Probability
Theory.
2021-11-25 (czwartek), 12:15 - 14:00
dr Piotr Dyszewski (Uniwersytet Wrocławski)
Limit laws for depths and heights of random trees
Streszczenie:
We will revisit the random split-trees introduced by Devroye "Universal limit laws for depths in random trees." SIAM Journal on Computing 28, no. 2 (1998): 409-432, and show how one can use an embedding into a continuous setting to present a new treatment of the model. We will illustrate our approach by giving a new proof of the central limit theorem for the depth of a split-tree. Moreover we will give a new result in the form of a limit theorem for the height of a split-tree. The talk is based on a joint work in progress with C. Mailer (University of Bath).
2021-10-21 (czwartek), 12:15 - 14:00
prof. Krzysztof Dębicki (Uniwersytet Wrocławski)
Ekstrema pól gaussowskich o nieaddytywnej strukturze zależności
Streszczenie:
Zanallizujemy dokładną asymptotykę ogona rozkładu
supremum dla klasy pól gaussowskich z niestandardową strukturą
zależności. Wyniki zostaną zilustrowane dwoma przykładami:
- asymptotycznymi własnościami "performance table" i związanymi z nią zagadnieniami własności wartości własnych macierzy GUE;
- asymptotykami procesów "chi".
Wystąpienie oparte jest na wspólnej pracy z L. Bai (Nankai Univ.) and P.Liu (Univ. of Essex).
2021-06-17 (czwartek), 12:15 - 14:00
Pavel Levlev (Université de Lausanne)
Proportional Parisian Reinsurance with many-fBm inputs
Streszczenie:
Consider the proportional reinsurance process with two (or more) companies sharing one risk process, modeled by large number of independent fractional Brownian motions. There are recent results on classical, joint and “at least one” ruin for this process, whereas the Parisian ruin seems to have not been studied before. In the talk I shall address the exact first order asymptotics in this case, which is motivated by a recent paper “Proportional reinsurance for fractional Brownian risk model” by Krzysztof Kępczyński.
2021-06-10 (czwartek), 12:15 - 14:00
Nikolai Kriukov (Université de Lausanne)
Extensions of Korshunov inequality for multidimensional processes
Streszczenie:
This talk is focused on the generalization of Korshunov inequality, which is helpful in simplifying the proof of double sum negligibility in ruin problems. It provides a non-asymptotic upper bound for the classical ruin probability of Brownian motion, which turns out to be asymptotically precise. The presenting results extend this inequality for the case of hitting some specific set and non-centered processes. In particular, we discuss applications to k-th simultaneous ruin probabilities.
2021-05-20 (czwartek), 12:15 - 14:00
prof. Matthias Meiners (Justus-Liebig-Universität Gießen)
Fluctuations of Biggins' martingales at complex parameters
Streszczenie:
The long-term behavior of a supercritical branching random walk
can be described and analyzed with the help of Biggins'
martingales, parametrized by real or complex numbers.
If certain sufficient conditions for the
convergence of the martingales to non-degenerate limits hold,
there are three different regimes of fluctuations of Biggins' martingales around their limits.
First, for parameters with small absolute values, the fluctuations are
Gaussian and the limit laws are scale mixtures of the real or
complex standard normal laws.
Second, there is a region in the parameter space in which
the martingale fluctuations are determined by the extremal
positions in the branching random walk. Finally, there is a
critical region (typically on the boundary of the set of
parameters for which the martingales converge to a non-degenerate
limit) where the fluctuations are stable-like and the limit laws
are the laws of randomly stopped Lévy processes satisfying
invariance properties similar to stability.
In my talk, I will present the fluctuations in all three regions
in the special case of a branching random walk
with binary splitting and independent Gaussian increments.
2021-05-06 (czwartek), 12:15 - 14:00
prof. Krzysztof Dębicki (Uniwersytet Wrocławski)
Czas pobytu pól losowych nad wysoką barierą
Streszczenie:
Dokonamy asymptotycznej analizy czasu przebywania pola losowego powyżej wysokiej bariery.
Przy dość ogólnych założeniach pokażemy, że istnieje interesujący związek między asymptotykami ogona czasu przebywania powyżej bariery i supremum. Ogólną teorię zilustrujemy przykładami.
Wystąpienie oparte jest na wspólnej pracy z E. Hashorvą, P. Liu oraz Z. Michną.
2021-04-22 (czwartek), 12:15 - 14:00
Grigori Jasnovidov (Université de Lausanne)
Ruin probability for Discrete and Continuous Gaussian Risk Models
Streszczenie:
In this presentation we focus on computation of the asymptotics of the ruin probabilities of Gaussian processes under the discrete time setup. We also study some properties of the Pickands constants appearing in the asymptotics.
2021-04-15 (czwartek), 12:15 - 14:00
dr Piotr Dyszewski (Technische Universität München)
The largest and smallest fragment in a k-regular self-similar fragmentation
Streszczenie:
We study the asymptotics of the $k$-regular self-similar fragmentation process. For $\alpha \geq 0$ and an integer $k\geq 2$, this is the Markov process $(I_t)_{t\geq 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$, and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally sized pieces at rate $u^\alpha$. Let $k^{-m_t}$ and $k^{-M_t}$ be the respective sizes of the largest and smallest fragments in $I_t$. By relating $(I_t)_{\geq 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t-g(t)|\leq 1$ and $|M_t-h(t)|\leq 1$ for all sufficiently large $t$. The talk is based on a joint work with Nina Gantert, Samuel G. G. Johnston, Joscha Prochno and Dominik Schmid.
2021-04-08 (czwartek), 12:15 - 14:00
dr Alexander Marynych (Taras Shevchenko National University of Kyiv)
Weak convergence of random geometric objects in non-Euclidean spaces
Streszczenie:
The analysis of conventional convex hulls of random points sampled from the uniform distribution on a compact convex subset
of the d-dimensional Euclidean space is the classical topic in stochastic geometry. In the past years there has been a splash of
activity around various generalized concepts of convexity both in geometric and probabilistic communities. The talk is devoted
to the discussion of two particular models of this kind. In the first model we consider a sample picked from the uniform distribution
on the upper semi-sphere and analyze its spherical convex hull. In the second model the sample is taken from the uniform distribution
in a convex body K in the d-dimensional Euclidean space and we focus on the analysis of its K-convex hull, that is, intersection of
all affine translates of K which contain the sample. Considered from an appropriate viewpoint, these two models incorporating
generalized notion of convexity exhibit a similar behavior which turns out to be very different from such in the classical setting.
2021-04-01 (czwartek), 12:15 - 14:00
prof. Dariusz Buraczewski (Uniwersytet Wrocławski)
Gałązkowy spacer losowy
Streszczenie:
Podczas wykładu opowiem o gałązkowym spacerze losowym i przedstawię podstawowe własności
dotyczące zachowania jego maksimum. W opisie tego procesu kluczową rolę odgrywa tzw. martyngał
pochodny. Opowiem o nowych wynikach otrzymanych wspólnie z Aleksandrem Iksanovem (Kijów)
oraz Bastienem Mallein (Paryż).
2021-03-25 (czwartek), 12:15 - 14:00
dr Ayan Bhattacharya (Indian Institute of Technology)
Extreme positions of regularly varying branching random walk in random environment
Streszczenie:
In this article, we consider a Branching Random Walk (BRW) on the real line where the underlying genealogical structure is given through a supercritical branching process in i.i.d. environment and satisfies Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (depends on the expected size of the n-th generation given the environment) maximum among positions at the n-th generation converges weakly to a scale-mixture of Frechét random variable. Furthermore, we derive the weak limit of the extremal processes composed of appropriately scaled positions at the n-th generation and show that the limit point process is a member of the randomly scaled scale-decorated Poisson point processes (SScDPPP). Hence, an analog of the predictions by Brunet and Derrida (2011) holds. This is a joint work with Zbigniew Palmowski.
2021-03-18 (czwartek), 12:15 - 14:00
prof. Zbigniew Palmowski (Politechnika Wrocławska)
On the renewal theorem for maxima on trees
Streszczenie:
We consider the distributional fixed-point equation:
\[R \stackrel{\mathcal{D}}{=} Q \vee \left( \bigvee_{i=1}^N C_i R_i \right),
\]
where the $\{R_i\}$ are i.i.d.~copies of $R$, independent of the vector
$(Q, N, \{C_i\})$, where $N \in \mathbb{N}$, $Q, \{C_i\} \geq 0$ and
$P(Q > 0) >0$. By setting $W = \log R$, $X_i = \log C_i$, $Y = \log Q$
it is equivalent to the high-order Lindley equation
\[W \stackrel{\mathcal{D}}{=} \max\left\{ Y, \, \max_{1 \leq i \leq N}
(X_i + W_i) \right\}.\]
It is known that under Kesten assumptions,
\[P(W > t) \sim H e^{-\alpha t}, \qquad t \to \infty,\]
where $\alpha>0$ solves the Cram\'er-Lundberg equation
$E \left[ \sum_{j=1}^N C_i ^\alpha \right] = E\left[ \sum_{i=1}^N
e^{\alpha X_i} \right] = 1$.
The main goal of this paper is to provide an explicit representation for
$P(W > t)$, which can be directly connected to the underlying weighted
branching process where $W$ is constructed and that can be used to
construct unbiased and strongly efficient estimators for all $t$.
Furthermore, we show how this new representation can be directly
analyzed using Alsmeyer's Markov renewal theorem, yielding an
alternative representation for the constant $H$. We provide numerical
examples illustrating the use of this new algorithm.
This is a joint work with Bojan Basrak, Michael Conroy and Mariana
Olvera-Cravioto.
2021-03-11 (czwartek), 12:15 - 14:00
prof. Alexander Iksanov (Taras Shevchenko National University of Kyiv)
On nested occupancy scheme in random environment
Streszczenie:
A nested occupancy scheme in random environment is a generalization of the Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. We say that the boxes belong to the jth level provided that their hitting probabilities are given by the j-fold fragmentation. Assuming that the number of balls is n, we shall present functional limit theorems for the number of occupied boxes in the jth level in two different settings: 1) j is fixed; 2) $j=j(n)$ diverges to infinity and $j(n)=o((\log n)^{1/2})$ as n tends to infinity.
The talk is based on recent joint works with D. Buraczewski, B. Dovgay, A. Gnedin, A.Marynych and I.Samoilenko.
2021-02-25 (czwartek), 12:15 - 14:00
dr Konrad Kolesko (Uniwersytet Wrocławski)
Twierdzenia graniczne dla procesów gałązkowych
Streszczenie:
Dla gałązkowego spaceru losowego $\{S(u)\}_{u \in \mathbb T} $ o przyrostach dodatnich oraz funkcji $\phi:[0,\infty) \mapsto \mathbb R$ definiujemy proces Crumpa-Mode’a-Jagersa $Z^\phi_t$ wzorem:
$$Z^\phi_t:=\sum_{u\in\mathbb T}\phi(t-S(u)).$$
W swojej pracy O. Nerman (1981) udowodnił, że przy pewnych naturalnych założeniach istnieje parametr $\alpha>0$, taki że
$$e^{-\alpha t} Z^\phi_t \stackrel{t\to\infty}{\longrightarrow} \int \phi(s)e^{-\alpha s}ds \cdot W\quad \text{p.n.}$$
dla pewnej zmiennej losowej $W$.
Podczas wykładu zaprezentuję twierdzenia graniczne dla procesu $Z^\phi_t$. Przedstawione wyniki oparte są na wspólnej pracy z Matthiasem Meinersem oraz Aleksandrem Iksanovem.