Borovkov meeting 2022

	Wednesday, 24 August	Thursday, 25 August	Friday, 26 August
12:00 — 12:40	Smorodina Natalia О ядрах некоторых случайных операторов Пусть $\xi_x(t)$ -- решение стохастического дифференциального уравнения $$ d\xi_x(t)=b(\xi_x(t))b^\prime(\xi_x(t))\,dt+b(\xi_x(t))\,dw(t),\ \ \ \ \xi_x(0)=x. $$ В пространстве $L_2(\mathbb{R})$ рассмотрим самосопряженный оператор $$\mathcal{A}=-\frac{1}{2}\,\frac{d}{dx}\big(b^2(x)\frac{d}{dx}\big)+V(x),$$ заданный на области определения $W_2^2(\mathbb{R})$. Относительно функций $b(x),V(x)$ мы будем предполагать выполнение следующих условий: 1. $V\in L_1(\mathbb{R}).$ 2. $b\in C_b^2$ и отделена от нуля. 3. Существует $b_0>0$ такое что $\underset{x\to\pm\infty}\lim b(x)=b_0.$ 4. $\underset{x\to\pm\infty}\lim b^\prime(x)=\underset{x\to\pm\infty}\lim b^{\prime\prime}(x)=0.$ 5. $\int_\mathbb{R}x^2(\|b(x)-b_0\|+\|b^{\prime}(x)\|)\,dx<\infty$. Из условий 1-5 вытекает, что спектр оператора $\mathcal{A}$ состоит из интервала $[0,\infty)$ и, возможно, нескольких отрицательных однократных собственных значений. Через $H_{a}\subset L_2(\mathbb{R})$ обозначим абсолютно непрерывное подпространство оператора $\mathcal{A}$, а через $P_{a}$ -- ортогональный проектор в $L_2(\mathbb{R})$ на $H_{a}$. Через $\mathcal{A}_0=\mathcal{A}P_{a}$ обозначим сужение оператора $\mathcal{A}$ на $H_{a}$. Для каждого $\lambda$, удовлетворяющего условию $\mathrm{Re}\,\lambda\leqslant 0$ определим случайный оператор $\mathcal{R}_\lambda^t$, полагая $$ \mathcal{R}_\lambda^tf(x)=\int_0^t e^{\lambda\tau}(P_{a}f)(\xi_x(\tau))e^{-\int_0^\tau V(\xi_x(s))\,ds} \,d\tau. $$ \textbf{Теорема 1.} 1. С вероятностью 1 оператор $\mathcal{R}_\lambda^t$ является ограниченным интегральным оператором в $L_2(\mathbb{R})$ вида $$\mathcal{R}_\lambda^tf(x)=\int_\mathbb{R}r_\lambda(t,x,y)f(y)\,dy,$$ причем при $\mathrm{Re}\,\lambda<0$ последнее равенство справедливо также для $t=\infty$. 2. Для любых $\lambda,t,x$ функция $r_\lambda(t,x,\cdot)\in W_2^\alpha$ для любого $\alpha\in[0,\frac{1}{2})$. \textbf{Теорема 2.} 1. Если $\mathrm{Re}\,\lambda< 0$ то для любого $f\in H_{a}$ выполнено \begin{equation} \mathbb{E}\int_\mathbb{R}r_\lambda(\infty,\cdot,y)f(y)\,dy=(\mathcal{A}_0-\lambda I)^{-1}f. \label{eq65} \end{equation} 2. Если $\mathrm{Re}\,\lambda= 0$ и $\lambda\neq 0$ то для любого $f\in H_{a}$ выполнено \begin{equation} \lim_{t\to\infty}\mathbb{E}\int_\mathbb{R}r_\lambda(t,\cdot,y)f(y)\,dy=(\mathcal{A}_0-\lambda I)^{-1}f. \label{eq70} \end{equation} При $\lambda=0$ равенство (\ref{eq70}) выполнено для любого $f\in \mathcal{D}(\mathcal{A}_0-\lambda I)^{-1}$.	Kovalevsky Artem Joint asymptotics of forward and backward processes of numbers of non-empty urns in infinite urn schemes We study the joint asymptotics of forward and backward processes of numbers of non-empty urns in an infinite urn scheme. The probabilities of balls hitting the urns are assumed to satisfy the conditions of regular decrease. We prove weak convergence to a two-dimensional Gaussian process. Its covariance function depends only on exponent of regular decrease of probabilities. The corollary of the main theorem asserts the weak convergence of the integral of the difference of forward and backward processes to the normal distribution. We obtain parameter estimates that have a joint normal distribution together with forward and backward processes. We use these estimates to construct statistical tests for the homogeneity of the urn scheme on the number of thrown balls. We analyse the statistical tests by simulation and apply them to the analysis of the homogeneity of texts in natural language
12:45 — 13:15	Lotov Vladimir, Tarasenko Anton Inequalities for the characteristics of the CUSUM procedure in a change point problem Получены оценки в виде неравенств для среднего времени задержки с реагированием на наличие разладки и для среднего времени до ложной тревоги при обнаружении разладки с помощью CUSUM процедуры.	Topchii Valentin Критические ветвящиеся процессы со счетным числом типов частиц и случайные графы. Рассмотрены генеалогические деревья ветвящихся процессов Гальтона-Ватсона. Изучается критический случай, соответствующий одновершинному случайному дереву с независимым одинаково распределенным количеством ребер для всех вершин. Среднее количество ребер, выходящих из вершины более низкого уровня равно 1. Одной из основополагающих теорем для данных процессов является теорема Яглома, утверждающая, что не вырождающиеся к далекому моменту времени n процессы содержат в данный момент времени количество частиц равное этому времени n, умноженному на экспоненциально распределенную случайную величину. Эти условные процессы удобно описывать в терминах редуцированных деревьев, которые получаются из генеалогических деревьев путем исключения поддеревьев, не доходящих до уровня n. Более сложная модель ветвящихся процессов Гальтона-Ватсона со счетным числом типов частиц, у которых типы потомков получаются суммированием типа родителя с независимыми одинаково распределенными многомерными случайными величинами можно представить как определенные выше в одномерном случае деревья с весами ребер и вершин. Описаны средние и дисперсии ряда характеристик редуцированных деревьев с весом, включая суммарный вес всех вершин на фиксированном уровне. Доказан ряд предельных теорем для редуцированных деревьев. Работа выполнена в рамках государственного задания ИМ СО РАН, проект FWNF-2022-0003.	Prokopenko Evgeny nan nan
13:15 — 13:30	Coffee-break
13:30 — 14:10	Korshunov Dmitry Large deviations for asymptotically space homogeneous Markov chains in two dimensions We discuss Markov chains in positive lattice quadrant whose transition probabilities converge at infinity. Assuming positive recurrence of the chain we study large deviations for its invariant probabilities under Cramer type conditions on jumps.	Tesemnikov Pavel, Foss Sergey Upper and lower bounds for the tail probabilities in a branching random walk with heavy-tailed distributions of jumps Let $ \{\xi_{i,j}\}_{i,j\ge 1} $ be a family of independent random variables (r.v.) with common distribution $ F $. We assume that $ F $ is centered, i.e. \begin{align} \mathbb{E} \xi_{1,1} = 0, \end{align} and heavy-tailed, i.e. \begin{align} \mathbb{E} e^{\lambda \xi_{1,1}} \equiv \int_{-\infty}^{\infty} e^{\lambda t} F(dt) = \infty \end{align} for all $ \lambda > 0 $. Define a family of random walks $ S_{i,n} $ as follows: \begin{align} S_{i, 0} = 0, \qquad S_{i,n} = \sum_{j=1}^{n} \xi_{i, j} \text{ for } n \ge 1. \end{align} Let $ Z $ be a positive integer-valued r.v. We study the tail distributional asymptotics for the following supremum: \begin{align} R_{\mu, Z}^{g} = \max_{1 \le i \le Z} \max_{0 \le n \le \mu} ( S_{i, n} - g(n)), \end{align} where $ \mu \le \infty $ is an arbitrary r.v. and $ g $ an arbitrary nonnegative function tending to infinity as $ n \to \infty $. We propose conditions under which the lower bound \begin{align} \mathbb{P} \left( R_{\mu, Z}^{g} > x \right) \ge (1 + o(1)) H_{\mu,Z}^{g}(x) \end{align} and the upper bound \begin{align} \mathbb{P} \left( R_{\mu, Z}^{g} > x \right) \le (1 + o(1)) H_{\mu,Z}^{g}(x) \end{align} hold with uniformity over all suitable random time instances $ \mu $ and functions $ g $. Here \begin{align} H_{\mu, Z}^{g}(x) = \sum_{n=1}^{\infty} \mathbb{E} \left[ Z \mathbb{I} (\mu \ge n) \right] \overline{F} (x + g(n)). \end{align} Note that the model under consideration is a particular case of a branching random walk having branching only in the first generation. \textbf{Acknowledgements:} The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2022-282 with the Ministry of Science and Higher Education of the Russian Federation.	Sakhanenko Alexander On asymptotics of the probability for a random process to stay above a moving boundary Let $X_1,X_2,\ldots$ be independent random variables. We always assume that the random walk $ S_n:=X_1+\ldots+X_n,\ n=1,2,\dots, $ belongs to the domain of attraction of the normal distribution: i.e. there exists an increasing to infinity sequence $\{b_n>0\}$ such that ${S_n}/{b_n}$ converges in distribution towards the standard normal law as $n\to\infty$. %For a non-random sequence $\{g_n=o(b_n)\}$ l Let $ T:=\inf\{k\geq1:S_k\leq g_k\} $ be the first crossing over the moving boundary $\{g_n=o(b_n)\}$ by the random walk $\{S_n\}$. We consider in the talk the asymptotic behavior of the upper tail $\mathbf{P}(T>n)$. %distributions of first-passage times over moving boundaries %\begin{gather} % \label{i4} %\mathbf{P}(T>n)=\mathbf{P}\Big(\min_{1\le k\le n}(S_k-g_k)>0\Big). %\end{gather} The known classical case is when random walks have zero means, finite variances and $ B_n^2:=\mathbf{E}[S_n^2]\to\infty. $ If the Lindeberg condition is satisfied then $$ \mathbf{P}(T>n)\sim\sqrt{\frac{2}{\pi}}\frac{U_n}{B_n} \quad \text{with}\quad U_n:=\mathbf{E}[S_n-g_n;T_g>n]. \eqno(1)$$ (See \emph{Ann. Probab.}, 2018, pp. 3313-3350.) In the present talk we focus on the further results in this direction. In particular, we are not going to assume that all summands have finite variances or even finite expectations. Denote by $X_n^{[u_n]}$ the truncation of the random variable $X_n$ on the levels $\pm u_n$, where $u_n/b_n\to0$ sufficiently slow. In this case $$ \mathbf{P}(T>n)\sim\sqrt{\frac{2}{\pi}}\frac{U_n(u_n)}{b_n} +J_n(u_n,b_n), \eqno(2)$$ where $U_n(u_n)$ is defined similar to $U_n$ in (1), but for the random walk $X_1^{[u_n]}+\dots+X_n^{[u_n]}$ instead of $S_n$. Note that the value $J_n(u_n,b_n)$ from (2) is found in explicit way as a function of distributions of positive jumps of random variables $X_1-u_n,\dots,X_n-u_n$. The talk is based on the joint works with D. Denisov and V. Wachtel. The research was funded by RFBR and DFG according to the research project \textnumero 20-51-12007.
14:10 — 16:00	Lunch
16:00 — 16:40	Mini-talks Prasolov Timofei Moments of the first descending epoch for a random walk with negative drift Bratenkov Miron Изменение скорости сходимости алгоритмов реконструкции изображений в диагностической ядерной медицине Efremov Egor Moderate deviations principle for m-dependent random variables in sublinear expectation space Ishkov Roman Research of the infection model in queues M/M/ k /0 Kusnatdinov Timur The greedy walk on real line Rezler Alexander Stability and instability of a random multiple access system with an energy harvesting mechanism Smirnov Ivan Probability approach for guessing game in random environment Tsybulskiy Dmitry Distribution of the length and the height of the regeneration cycle for a random walk with drift Shelepova Anastasiya Асимптотика распределения момента выхода за невозрастающую границу для обобщенных процессов восстановления. Lukyanov Andrey Метод двойного бутстрапа для оценивания степенного индекса по экспектилям. Trushin Alexander Challenges in genome-wide association studies Tesemnikov Pavel On the distribution of the length of the shortest path in a generalised Barak-Erd\H{o}s graph	Shemyakin Arkady Hellinger Information Matrix in Parametric Estimation and Objective Priors Hellinger information as a local characteristic of parametric distribution families was first introduced in (Shemyakin, 1992). It is related to the definition of Hellinger distance between two parametric values. Under certain regularity conditions, local behavior of the Hellinger distance is closely related to Fisher information and the geometry of Riemann manifolds. Nonregular distributions (non-differentiable distribution densities or undefined Fisher information), including uniform, require using analogues or extensions of Fisher information. Hellinger information may serve to construct information inequalities of Cramer-Rao type, extending the lower bounds of the Bayes risk (Borovkov and Sakhanienko, 1980) to the nonregular case (Shemyakin, 1991). A construction of objective or non-informative priors based on Hellinger information was suggested in Shemyakin (2014). Hellinger priors extend the Jeffreys’ rule to nonregular cases. For many examples, they are identical or close to the reference priors (Berger, Bernardo and Sun, 2009) or probability matching priors (Ghosal and Samanta, 1997). Most of the paper was dedicated to one-dimensional case, but the matrix definition of Hellinger information was also introduced for higher dimensions. Conditions of existence and nonnegative definite property of Hellinger information matrix were not discussed. Hellinger information was also applied by Lin, Martin, and Yang (2019) to problems of optimal experimental design. A special class of parametric problems was considered, requiring directional definition of Hellinger information, but not a full construction of Hellinger information matrix. In the present paper, a general definition, existence and nonnegative definite property of Hellinger information matrix is considered for nonregular settings described in Ibragimov and Has’minskii (1981).	Wachtel Vitali Asymptotic expansions for first-passage times of an oscillating random walk In this talk I shall consider asymptotic expansions for the tail of the distribution of the time when an oscillating random walk crosses a fixed level $-x\le 0$ for the first time . Furthermore, I shall discuss a connection between such expansions and polyharmonic functions for killed random walks.
16:45 — 17:15	Borovkov Konstantin Parisian ruin with random deficit-dependent delays for spectrally negative L´evy processes We consider an interesting natural extension to the Parisian ruin problem under the assumption that the risk reserve dynamics are given by a spectrally negative L´evy process. The distinctive feature of this extension is that the distribution of the random implementation delay windows’ lengths can depend on the deficit at the epochs when the risk reserve process turns negative, starting a new negative excursion. This includes the possibility of an immediate ruin when the deficit hits a certain subset. In this general setting, we derive a closed-from expression for the Parisian ruin probability and the joint Laplace transform of the Parisian ruin time and the deficit at ruin. [Joint work with Duy Phat Nguyen.]	Logachev Artem, Mogulskii Anatolii Принципы умеренно больших уклонений для траекторий неоднородных случайных блужданий В докладе будет рассмотрена нормированная ломанная построенная по суммам независимых, вообще говоря, разнораспределенных случайных величин. При различных моментных условиях на случайные величины будут изложены теоремы, содержащие принципы умеренно больших уклонений для таких ломанных в пространстве непрерывных на отрезке [0,1] функций. Также будет указана связь между зоной, в которой выполнен принцип умеренно больших уклонений и тем моментом, который существует у случайных величин.
17:15 — 17:30	Welcome party + стенды	Coffee-break
17:30 — 18:10	Zuyev Sergei Probing harmony with algebra (at least, statistically) In a recent statistical study, US researches quantified attractiveness of a face by using measures of deviation from canonical "standards" like equality of eye width to interocular distance or golden ratio of nose to chin distance to nose width. The actual attractiveness formula is kept as a commercial secret, but using available published data we shall discuss if attractiveness is really a function of the geometry of a face and to which extent the harmony can be described by the algebra (even statistically). In the course of the talk we shall discuss the latest scientific results in the intersection of physiology, psychology, statistics and computer graphics on what attractiveness is and its biological roots.	Rybko Alexander Dynamic systems related to the emergence of alpha - rhythm of brain cortex The following class of dynamic systems is studied: N points are rotating clockwise with speed equal 1 on a unit circle. The connected oriented graph F with N nodes is given. The (unnown) real function on the circle f(x) is given.There is a picked point 0 on the circle where f(0)=0. The rotating points are making jumps as well: at moment t when any rotated point n = 1,...,N reaches 0, then each point m neighbouring by graph F to point n, jumps to the distance of f(m(t)) on the circle.The function f(x) depends on N and the graph F is random. It is clear that these dynamic systems have trivial invariant state when all the N points merging in one big atom rotating on the circle (without any jumps as f(0)=0). Usually nontrivial states exist also for such dynamic systems. The problem is to find such a natural function f(x) for which we shall converge to this trivial invariant state with high probability when t tends to infinity for growing N.
18:15 — 18:45	Denisov Denis Local probabilities for asymptotically stable random walks in half space We consider an asymptotically stable multidimensional random walk $S(n)=(S_1(n),\ldots, S_d(n) )$. Let $\tau_x:=\min\{n>0: x_{1}+S_1(n)\le 0\}$ be the first time the random walk $x+S(n)$ leaves the upper half-space. We study the asymptotics of $p_n(x,y):= \mathbb{P}(x+S(n) \in y+\Delta, \tau_x>n)$ as $n$ tends to infinity, where $\Delta$ is a fixed cube. We obtain exact asymptotics in the regime of normal and small deviations and obtain accurate bounds in the regime of large deviations. From that we obtain the local asymptotics for the Green function of $G(x,y):=\sum_n p_n(x,y)$, as $\|y\|$ and/or $\|x\|$ tend to infinity. This is joint work with V. Wachtel	Chebunin Mikhail Harris Ergodicity of a Split Transmission Control Protocol Additive-increase multiplicative-decrease transmission control protocols are well known and have been studied in numerous papers. It is much more difficult to study the properties of systems of interacting protocols. We consider a queueing system in which both the intensity of the input stream and the intensity of the service follow a TCP protocol and the dynamics of the latter depends on both intensities. This kind of stochastic system was proposed by Baccelli, Carofiglio, and Foss in 2009, who have proved the positive recurrence of the underlying Markov chain and studied a number of statistical properties of the model. In this paper, we introduce a more general stochastic model and prove a stronger statement: the Harris ergodicity of the corresponding Markov chain. (joint work with Sergey Foss)

Wednesday, 24 August

Thursday, 25 August

Friday, 26 August

12:00 — 12:40

Smorodina Natalia

О ядрах некоторых случайных операторов

Пусть $\xi_x(t)$ -- решение стохастического дифференциального уравнения $$ d\xi_x(t)=b(\xi_x(t))b^\prime(\xi_x(t))\,dt+b(\xi_x(t))\,dw(t),\ \ \ \ \xi_x(0)=x. $$ В пространстве $L_2(\mathbb{R})$ рассмотрим самосопряженный оператор $$\mathcal{A}=-\frac{1}{2}\,\frac{d}{dx}\big(b^2(x)\frac{d}{dx}\big)+V(x),$$ заданный на области определения $W_2^2(\mathbb{R})$. Относительно функций $b(x),V(x)$ мы будем предполагать выполнение следующих условий: 1. $V\in L_1(\mathbb{R}).$ 2. $b\in C_b^2$ и отделена от нуля. 3. Существует $b_0>0$ такое что $\underset{x\to\pm\infty}\lim b(x)=b_0.$ 4. $\underset{x\to\pm\infty}\lim b^\prime(x)=\underset{x\to\pm\infty}\lim b^{\prime\prime}(x)=0.$ 5. $\int_\mathbb{R}x^2(|b(x)-b_0|+|b^{\prime}(x)|)\,dx<\infty$. Из условий 1-5 вытекает, что спектр оператора $\mathcal{A}$ состоит из интервала $[0,\infty)$ и, возможно, нескольких отрицательных однократных собственных значений. Через $H_{a}\subset L_2(\mathbb{R})$ обозначим абсолютно непрерывное подпространство оператора $\mathcal{A}$, а через $P_{a}$ -- ортогональный проектор в $L_2(\mathbb{R})$ на $H_{a}$. Через $\mathcal{A}_0=\mathcal{A}P_{a}$ обозначим сужение оператора $\mathcal{A}$ на $H_{a}$. Для каждого $\lambda$, удовлетворяющего условию $\mathrm{Re}\,\lambda\leqslant 0$ определим случайный оператор $\mathcal{R}_\lambda^t$, полагая $$ \mathcal{R}_\lambda^tf(x)=\int_0^t e^{\lambda\tau}(P_{a}f)(\xi_x(\tau))e^{-\int_0^\tau V(\xi_x(s))\,ds} \,d\tau. $$ \textbf{Теорема 1.} 1. С вероятностью 1 оператор $\mathcal{R}_\lambda^t$ является ограниченным интегральным оператором в $L_2(\mathbb{R})$ вида $$\mathcal{R}_\lambda^tf(x)=\int_\mathbb{R}r_\lambda(t,x,y)f(y)\,dy,$$ причем при $\mathrm{Re}\,\lambda<0$ последнее равенство справедливо также для $t=\infty$. 2. Для любых $\lambda,t,x$ функция $r_\lambda(t,x,\cdot)\in W_2^\alpha$ для любого $\alpha\in[0,\frac{1}{2})$. \textbf{Теорема 2.} 1. Если $\mathrm{Re}\,\lambda< 0$ то для любого $f\in H_{a}$ выполнено \begin{equation} \mathbb{E}\int_\mathbb{R}r_\lambda(\infty,\cdot,y)f(y)\,dy=(\mathcal{A}_0-\lambda I)^{-1}f. \label{eq65} \end{equation} 2. Если $\mathrm{Re}\,\lambda= 0$ и $\lambda\neq 0$ то для любого $f\in H_{a}$ выполнено \begin{equation} \lim_{t\to\infty}\mathbb{E}\int_\mathbb{R}r_\lambda(t,\cdot,y)f(y)\,dy=(\mathcal{A}_0-\lambda I)^{-1}f. \label{eq70} \end{equation} При $\lambda=0$ равенство (\ref{eq70}) выполнено для любого $f\in \mathcal{D}(\mathcal{A}_0-\lambda I)^{-1}$.

Kovalevsky Artem

Joint asymptotics of forward and backward processes of numbers of non-empty urns in infinite urn schemes

We study the joint asymptotics of forward and backward processes of numbers of non-empty urns in an infinite urn scheme. The probabilities of balls hitting the urns are assumed to satisfy the conditions of regular decrease. We prove weak convergence to a two-dimensional Gaussian process. Its covariance function depends only on exponent of regular decrease of probabilities. The corollary of the main theorem asserts the weak convergence of the integral of the difference of forward and backward processes to the normal distribution. We obtain parameter estimates that have a joint normal distribution together with forward and backward processes. We use these estimates to construct statistical tests for the homogeneity of the urn scheme on the number of thrown balls. We analyse the statistical tests by simulation and apply them to the analysis of the homogeneity of texts in natural language

12:45 — 13:15

Lotov Vladimir, Tarasenko Anton

Inequalities for the characteristics of the CUSUM procedure in a change point problem

Получены оценки в виде неравенств для среднего времени задержки с реагированием на наличие разладки и для среднего времени до ложной тревоги при обнаружении разладки с помощью CUSUM процедуры.

Topchii Valentin

Критические ветвящиеся процессы со счетным числом типов частиц и случайные графы.

Рассмотрены генеалогические деревья ветвящихся процессов Гальтона-Ватсона. Изучается критический случай, соответствующий одновершинному случайному дереву с независимым одинаково распределенным количеством ребер для всех вершин. Среднее количество ребер, выходящих из вершины более низкого уровня равно 1. Одной из основополагающих теорем для данных процессов является теорема Яглома, утверждающая, что не вырождающиеся к далекому моменту времени n процессы содержат в данный момент времени количество частиц равное этому времени n, умноженному на экспоненциально распределенную случайную величину. Эти условные процессы удобно описывать в терминах редуцированных деревьев, которые получаются из генеалогических деревьев путем исключения поддеревьев, не доходящих до уровня n. Более сложная модель ветвящихся процессов Гальтона-Ватсона со счетным числом типов частиц, у которых типы потомков получаются суммированием типа родителя с независимыми одинаково распределенными многомерными случайными величинами можно представить как определенные выше в одномерном случае деревья с весами ребер и вершин. Описаны средние и дисперсии ряда характеристик редуцированных деревьев с весом, включая суммарный вес всех вершин на фиксированном уровне. Доказан ряд предельных теорем для редуцированных деревьев. Работа выполнена в рамках государственного задания ИМ СО РАН, проект FWNF-2022-0003.

Prokopenko Evgeny

nan

13:15 — 13:30

Coffee-break

13:30 — 14:10

Korshunov Dmitry

Large deviations for asymptotically space homogeneous Markov chains in two dimensions

We discuss Markov chains in positive lattice quadrant whose transition probabilities converge at infinity. Assuming positive recurrence of the chain we study large deviations for its invariant probabilities under Cramer type conditions on jumps.

Tesemnikov Pavel, Foss Sergey

Upper and lower bounds for the tail probabilities in a branching random walk with heavy-tailed distributions of jumps

Let $ \{\xi_{i,j}\}_{i,j\ge 1} $ be a family of independent random variables (r.v.) with common distribution $ F $. We assume that $ F $ is centered, i.e. \begin{align*} \mathbb{E} \xi_{1,1} = 0, \end{align*} and heavy-tailed, i.e. \begin{align*} \mathbb{E} e^{\lambda \xi_{1,1}} \equiv \int_{-\infty}^{\infty} e^{\lambda t} F(dt) = \infty \end{align*} for all $ \lambda > 0 $. Define a family of random walks $ S_{i,n} $ as follows: \begin{align*} S_{i, 0} = 0, \qquad S_{i,n} = \sum_{j=1}^{n} \xi_{i, j} \text{ for } n \ge 1. \end{align*} Let $ Z $ be a positive integer-valued r.v. We study the tail distributional asymptotics for the following supremum: \begin{align*} R_{\mu, Z}^{g} = \max_{1 \le i \le Z} \max_{0 \le n \le \mu} ( S_{i, n} - g(n)), \end{align*} where $ \mu \le \infty $ is an arbitrary r.v. and $ g $ an arbitrary nonnegative function tending to infinity as $ n \to \infty $. We propose conditions under which the lower bound \begin{align*} \mathbb{P} \left( R_{\mu, Z}^{g} > x \right) \ge (1 + o(1)) H_{\mu,Z}^{g}(x) \end{align*} and the upper bound \begin{align*} \mathbb{P} \left( R_{\mu, Z}^{g} > x \right) \le (1 + o(1)) H_{\mu,Z}^{g}(x) \end{align*} hold with uniformity over all suitable random time instances $ \mu $ and functions $ g $. Here \begin{align*} H_{\mu, Z}^{g}(x) = \sum_{n=1}^{\infty} \mathbb{E} \left[ Z \mathbb{I} (\mu \ge n) \right] \overline{F} (x + g(n)). \end{align*} Note that the model under consideration is a particular case of a branching random walk having branching only in the first generation. \textbf{Acknowledgements:} The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2022-282 with the Ministry of Science and Higher Education of the Russian Federation.

Sakhanenko Alexander

On asymptotics of the probability for a random process to stay above a moving boundary

Let $X_1,X_2,\ldots$ be independent random variables. We always assume that the random walk $ S_n:=X_1+\ldots+X_n,\ n=1,2,\dots, $ belongs to the domain of attraction of the normal distribution: i.e. there exists an increasing to infinity sequence $\{b_n>0\}$ such that ${S_n}/{b_n}$ converges in distribution towards the standard normal law as $n\to\infty$. %For a non-random sequence $\{g_n=o(b_n)\}$ l Let $ T:=\inf\{k\geq1:S_k\leq g_k\} $ be the first crossing over the moving boundary $\{g_n=o(b_n)\}$ by the random walk $\{S_n\}$. We consider in the talk the asymptotic behavior of the upper tail $\mathbf{P}(T>n)$. %distributions of first-passage times over moving boundaries %\begin{gather*} % \label{i4} %\mathbf{P}(T>n)=\mathbf{P}\Big(\min_{1\le k\le n}(S_k-g_k)>0\Big). %\end{gather*} The known classical case is when random walks have zero means, finite variances and $ B_n^2:=\mathbf{E}[S_n^2]\to\infty. $ If the Lindeberg condition is satisfied then $$ \mathbf{P}(T>n)\sim\sqrt{\frac{2}{\pi}}\frac{U_n}{B_n} \quad \text{with}\quad U_n:=\mathbf{E}[S_n-g_n;T_g>n]. \eqno(1)$$ (See \emph{Ann. Probab.}, 2018, pp. 3313-3350.) In the present talk we focus on the further results in this direction. In particular, we are not going to assume that all summands have finite variances or even finite expectations. Denote by $X_n^{[u_n]}$ the truncation of the random variable $X_n$ on the levels $\pm u_n$, where $u_n/b_n\to0$ sufficiently slow. In this case $$ \mathbf{P}(T>n)\sim\sqrt{\frac{2}{\pi}}\frac{U_n(u_n)}{b_n} +J_n(u_n,b_n), \eqno(2)$$ where $U_n(u_n)$ is defined similar to $U_n$ in (1), but for the random walk $X_1^{[u_n]}+\dots+X_n^{[u_n]}$ instead of $S_n$. Note that the value $J_n(u_n,b_n)$ from (2) is found in explicit way as a function of distributions of positive jumps of random variables $X_1-u_n,\dots,X_n-u_n$. The talk is based on the joint works with D. Denisov and V. Wachtel. The research was funded by RFBR and DFG according to the research project \textnumero 20-51-12007.

14:10 — 16:00

Lunch

16:00 — 16:40

Mini-talks

Prasolov Timofei

Moments of the first descending epoch for a random walk with negative drift

Bratenkov Miron

Изменение скорости сходимости алгоритмов реконструкции изображений в диагностической ядерной медицине

Efremov Egor

Moderate deviations principle for m-dependent random variables in sublinear expectation space

Ishkov Roman

Research of the infection model in queues M/M/ k /0

Kusnatdinov Timur

The greedy walk on real line

Rezler Alexander

Stability and instability of a random multiple access system with an energy harvesting mechanism

Smirnov Ivan

Probability approach for guessing game in random environment

Tsybulskiy Dmitry

Distribution of the length and the height of the regeneration cycle for a random walk with drift

Shelepova Anastasiya

Асимптотика распределения момента выхода за невозрастающую границу для обобщенных процессов восстановления.

Lukyanov Andrey

Метод двойного бутстрапа для оценивания степенного индекса по экспектилям.

Trushin Alexander

Challenges in genome-wide association studies

Tesemnikov Pavel

On the distribution of the length of the shortest path in a generalised Barak-Erd\H{o}s graph

Shemyakin Arkady

Hellinger Information Matrix in Parametric Estimation and Objective Priors

Hellinger information as a local characteristic of parametric distribution families was first introduced in (Shemyakin, 1992). It is related to the definition of Hellinger distance between two parametric values. Under certain regularity conditions, local behavior of the Hellinger distance is closely related to Fisher information and the geometry of Riemann manifolds. Nonregular distributions (non-differentiable distribution densities or undefined Fisher information), including uniform, require using analogues or extensions of Fisher information. Hellinger information may serve to construct information inequalities of Cramer-Rao type, extending the lower bounds of the Bayes risk (Borovkov and Sakhanienko, 1980) to the nonregular case (Shemyakin, 1991). A construction of objective or non-informative priors based on Hellinger information was suggested in Shemyakin (2014). Hellinger priors extend the Jeffreys’ rule to nonregular cases. For many examples, they are identical or close to the reference priors (Berger, Bernardo and Sun, 2009) or probability matching priors (Ghosal and Samanta, 1997). Most of the paper was dedicated to one-dimensional case, but the matrix definition of Hellinger information was also introduced for higher dimensions. Conditions of existence and nonnegative definite property of Hellinger information matrix were not discussed. Hellinger information was also applied by Lin, Martin, and Yang (2019) to problems of optimal experimental design. A special class of parametric problems was considered, requiring directional definition of Hellinger information, but not a full construction of Hellinger information matrix. In the present paper, a general definition, existence and nonnegative definite property of Hellinger information matrix is considered for nonregular settings described in Ibragimov and Has’minskii (1981).

Wachtel Vitali

Asymptotic expansions for first-passage times of an oscillating random walk

In this talk I shall consider asymptotic expansions for the tail of the distribution of the time when an oscillating random walk crosses a fixed level $-x\le 0$ for the first time . Furthermore, I shall discuss a connection between such expansions and polyharmonic functions for killed random walks.

16:45 — 17:15

Borovkov Konstantin

Parisian ruin with random deficit-dependent delays for spectrally negative L´evy processes

We consider an interesting natural extension to the Parisian ruin problem under the assumption that the risk reserve dynamics are given by a spectrally negative L´evy process. The distinctive feature of this extension is that the distribution of the random implementation delay windows’ lengths can depend on the deficit at the epochs when the risk reserve process turns negative, starting a new negative excursion. This includes the possibility of an immediate ruin when the deficit hits a certain subset. In this general setting, we derive a closed-from expression for the Parisian ruin probability and the joint Laplace transform of the Parisian ruin time and the deficit at ruin. [Joint work with Duy Phat Nguyen.]

Logachev Artem, Mogulskii Anatolii

Принципы умеренно больших уклонений для траекторий неоднородных случайных блужданий

В докладе будет рассмотрена нормированная ломанная построенная по суммам независимых, вообще говоря, разнораспределенных случайных величин. При различных моментных условиях на случайные величины будут изложены теоремы, содержащие принципы умеренно больших уклонений для таких ломанных в пространстве непрерывных на отрезке [0,1] функций. Также будет указана связь между зоной, в которой выполнен принцип умеренно больших уклонений и тем моментом, который существует у случайных величин.

17:15 — 17:30

Welcome party + стенды

Coffee-break

17:30 — 18:10

Zuyev Sergei

Probing harmony with algebra (at least, statistically)

In a recent statistical study, US researches quantified attractiveness of a face by using measures of deviation from canonical "standards" like equality of eye width to interocular distance or golden ratio of nose to chin distance to nose width. The actual attractiveness formula is kept as a commercial secret, but using available published data we shall discuss if attractiveness is really a function of the geometry of a face and to which extent the harmony can be described by the algebra (even statistically). In the course of the talk we shall discuss the latest scientific results in the intersection of physiology, psychology, statistics and computer graphics on what attractiveness is and its biological roots.

Rybko Alexander

Dynamic systems related to the emergence of alpha - rhythm of brain cortex

The following class of dynamic systems is studied: N points are rotating clockwise with speed equal 1 on a unit circle. The connected oriented graph F with N nodes is given. The (unnown) real function on the circle f(x) is given.There is a picked point 0 on the circle where f(0)=0. The rotating points are making jumps as well: at moment t when any rotated point n = 1,...,N reaches 0, then each point m neighbouring by graph F to point n, jumps to the distance of f(m(t)) on the circle.The function f(x) depends on N and the graph F is random. It is clear that these dynamic systems have trivial invariant state when all the N points merging in one big atom rotating on the circle (without any jumps as f(0)=0). Usually nontrivial states exist also for such dynamic systems. The problem is to find such a natural function f(x) for which we shall converge to this trivial invariant state with high probability when t tends to infinity for growing N.

18:15 — 18:45

Denisov Denis

Local probabilities for asymptotically stable random walks in half space

We consider an asymptotically stable multidimensional random walk $S(n)=(S_1(n),\ldots, S_d(n) )$. Let $\tau_x:=\min\{n>0: x_{1}+S_1(n)\le 0\}$ be the first time the random walk $x+S(n)$ leaves the upper half-space. We study the asymptotics of $p_n(x,y):= \mathbb{P}(x+S(n) \in y+\Delta, \tau_x>n)$ as $n$ tends to infinity, where $\Delta$ is a fixed cube. We obtain exact asymptotics in the regime of normal and small deviations and obtain accurate bounds in the regime of large deviations. From that we obtain the local asymptotics for the Green function of $G(x,y):=\sum_n p_n(x,y)$, as $|y|$ and/or $|x|$ tend to infinity. This is joint work with V. Wachtel

Chebunin Mikhail

Harris Ergodicity of a Split Transmission Control Protocol

Additive-increase multiplicative-decrease transmission control protocols are well known and have been studied in numerous papers. It is much more difficult to study the properties of systems of interacting protocols. We consider a queueing system in which both the intensity of the input stream and the intensity of the service follow a TCP protocol and the dynamics of the latter depends on both intensities. This kind of stochastic system was proposed by Baccelli, Carofiglio, and Foss in 2009, who have proved the positive recurrence of the underlying Markov chain and studied a number of statistical properties of the model. In this paper, we introduce a more general stochastic model and prove a stronger statement: the Harris ergodicity of the corresponding Markov chain. (joint work with Sergey Foss)