Borovkov meeting

	Wednesday, 24 August	Thursday, 25 August	Friday, 26 August
12:00 — 12:40	Smorodina Natalia Video, Slides On a Kernels of Some Random Operators Let $\xi_x(t)$ be a solution of the stochastic differential equation $$ d\xi_x(t)=b(\xi_x(t))b^\prime(\xi_x(t))\,dt+b(\xi_x(t))\,dw(t),\ \ \ \ \xi_x(0)=x. $$ In the space $L_2(\mathbb{R})$, consider the self-adjoint operator $$\mathcal{A}=-\frac{1}{2}\,\frac{d}{dx}\big(b^2(x)\frac{d}{dx}\big)+V( x),$$ defined on the domain $W_2^2(\mathbb{R})$. Regarding the functions $b(x),V(x)$, we will assume that the following conditions are satisfied: 1. $V\in L_1(\mathbb{R}).$ 2. $b\in C_b^2$ and separated from zero. 3. There exists $b_0>0$ such that $\underset{x\to\pm\infty}\lim b(x)=b_0.$ 4. $\underset{x\to\pm\infty}\lim b^\prime(x)=\underset{x\to\pm\infty}\lim b^{\prime\prime}(x)=0 .$ 5. $\int_\mathbb{R}x^2(\|b(x)-b_0\|+\|b^{\prime}(x)\|)\,dx<\infty$. The conditions 1-5 imply that the spectrum of the operator $\mathcal{A}$ consists of the interval $[0,\infty)$ and, possibly, several negative single eigenvalues. Denote by $H_{a}\subset L_2(\mathbb{R})$ an absolutely continuous subspace of the operator $\mathcal{A}$, and by $P_{a}$ the orthogonal projection into $L_2(\mathbb{R})$ on $H_{a}$. Let $\mathcal{A}_0=\mathcal{A}P_{a}$ denote the restriction of the operator $\mathcal{A}$ on $H_{a}$. For each $\lambda$ satisfying the condition $\mathrm{Re}\,\lambda\leqslant 0$, we define a random operator $\mathcal{R}_\lambda^t$ by setting $$\mathcal{R}_\lambda^tf(x)=\int_0^t e^{\lambda\tau}(P_{a}f)(\xi_x(\tau))e^{-\int_0^\tau V(\xi_x(s))\,ds} \,d\tau. $$ \textbf{Theorem 1.} 1. With probability 1 the operator $\mathcal{R}_\lambda^t$ is a bounded integral operator in $L_2(\mathbb{R})$ of the form $$\mathcal{R}_\lambda^tf(x)=\int_\mathbb{R}r_\lambda(t,x,y)f(y)\,dy,$$ where the last equality is also valid for $t=\infty$ in the case $\mathrm{Re }\,\lambda<0$. 2. For any $\lambda,t,x$ the function $r_\lambda(t,x,\cdot)\in W_2^\alpha$ for every $\alpha\in[0,\frac{1}{2})$. \textbf{Theorem 2.} 1. If $\mathrm{Re}\,\lambda< 0$ then for all $f\in H_{a}$ \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. If $\mathrm{Re}\,\lambda= 0$ and $\lambda\neq 0$ then for all $f\in H_{a}$ \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} For $\lambda=0$ the last equality (\ref{eq70}) holds for every $f\in \mathcal{D}(\mathcal{A}_0-\lambda I)^{-1}$. The work is supported by RSF, project \textnumero 22-21-00016.	Kovalevskii Artem Video, Slides 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 the 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 the 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.	Zaporozhets Dmitry Video Random Polynomials Having No Real Zeros In their 2002 paper, Dembo, Poonen, Shao and Zeitouni obtained a power asymptotic $n$ decreasing probability that a random polynomial of even degree $n$ with i.i.d. coefficients has no real zeros. The exact power exponent was not found, but it was conjectured that it is equal to $-3/4$. Only in the summer of 2021 FitzGerald, Tribe, and Zaboronski posted the work in the arXiv with its proof. In this talk we will consider a similar problem for random polynomials whose coefficients have binomial variance. These polynomials were first considered by Kostlan, Shub, and Smale in their works in the early 90s of the last century.
12:45 — 13:15	Tarasenko Anton, Lotov Vladimir Video, Slides Inequalities for the Characteristics of The Cusum Procedure in a Change Point Problem We obtain an upper bound for the average delay time with a response to the presence of a change point and for the average time to a false alarm when a change point is detected using the CUSUM procedure.	Topchii Valentin Video, Slides Critical Galton-Watson Branching Processes with a Countable Set of Particle Types and Random Graphs We consider genealogical trees of Galton-Watson branching processes and study the critical case corresponding to a one-vertex random tree with an independent, identically distributed number of edges for all vertices. The average number of edges coming out of a lower level vertex is $1$. One of the fundamental theorems for these processes is a Yaglom-type theorem, which states that processes that do not degenerate to a distant time $n$ contain at a given time the number of particles equal to this time $n$ times exponentially distributed random variable. It is convenient to describe these conditional processes in terms of reduced trees, which are obtained from genealogical trees by eliminating subtrees that do not reach level $n$. A more complex model of Galton-Watson branching processes with a countable set of particle types, in which the types of descendants are obtained by summing the parent type with independent identically distributed multidimensional random variables, can be represented as trees with weights of edges and vertices defined above in the one-dimensional case. We describe the averages and variances of a number of characteristics of reduced weighted trees, including the total weight of all vertices at a fixed level. We prove a number of limit theorems for reduced trees. The work was carried out within the framework of the state task of the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project FWNF-2022-0003.	Prokopenko Evgeny Video, Slides Multi-Normex approach for evaluating the sum of heavy tailed random vectors We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'Normex' approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named $d$-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles. This is joint work with Marie Kratz.
13:15 — 13:30	Coffee-break
13:30 — 14:10	Korshunov Dmitry Video Large Deviations for Asymptotically Space Homogeneous Markov Chains in Two Dimensions We discuss Markov chains in the 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 Video, Slides 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. 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 Video 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 $\{b_n\}$ such that ${S_n}/{b_n}$ converges in distribution towards the standard normal law as $n\to\infty$. Let $ T:=\inf\{k\geq1:S_k\leq g_k\} $ be the first crossing time 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 $\mathbf{P}(T>n)$. 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 Video Prasolov Timofei Moments of the First Descending Epoch for a Random Walk with Negative Drift Efremov Egor Moderate Deviations Principle for M-Dependent Random Variables in Sublinear Expectation Space Ishkov Roman Research of the Infection Model in Queues $M / M / k / 0$ 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 yhe Regeneration Cycle for a Random Walk with Drift Shelepova Anastasiya Asymptotics of the Distribution of the Exit Time Beyond a Non-Increasing Boundary for a Compound Renewal Process Lukyanov Andrey Double Bootstrap Method for Tail Index Estimation by Expectiles Trushin Alexander Mathematics 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 Slides 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 Sakhanenko, 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).	Chebunin Mikhail Video, Slides 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. This is joint work with Sergey Foss.
16:45 — 17:15	Borovkov Konstantin Slides Parisian Ruin with Random Deficit-Dependent Delays for Spectrally Negative L\'{e}vy 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\'{e}vy 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. This is joint work with Duy Phat Nguyen.	Logachev Artem, Mogulskii Anatolii Video, Slides Moderate Deviations Principles for Trajectories of Inhomogeneous Random Walks We consider a normalized piecewise linear curve constructed from sums of independent random variables that may have different distributions. Under various moment conditions on random variables we present theorems containing the principles of moderate large deviations for such piecewise linear curves in the space of continuous functions on the interval [0,1]. We also point out the connection between the zone in which the principle of moderately large deviations is fulfilled and the moment that exists for random variables.
17:15 — 17:30	Welcome party + стенды	Coffee-break
17:30 — 18:10	Wachtel Vitali Video, Slides 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.	Rybko Alexander Video 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 to $1$ 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 a trivial invariant state when all the $N$ points merge 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 Video, Slides 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.	Zuyev Sergei Video Probing Harmony with Algebra (at least, statistically) In a recent statistical study, US researchers 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.

Wednesday, 24 August

Thursday, 25 August

Friday, 26 August

12:00 — 12:40

Smorodina Natalia
Video, Slides

On a Kernels of Some Random Operators

Let $\xi_x(t)$ be a solution of the stochastic differential equation $$ d\xi_x(t)=b(\xi_x(t))b^\prime(\xi_x(t))\,dt+b(\xi_x(t))\,dw(t),\ \ \ \ \xi_x(0)=x. $$ In the space $L_2(\mathbb{R})$, consider the self-adjoint operator $$\mathcal{A}=-\frac{1}{2}\,\frac{d}{dx}\big(b^2(x)\frac{d}{dx}\big)+V( x),$$ defined on the domain $W_2^2(\mathbb{R})$. Regarding the functions $b(x),V(x)$, we will assume that the following conditions are satisfied: 1. $V\in L_1(\mathbb{R}).$ 2. $b\in C_b^2$ and separated from zero. 3. There exists $b_0>0$ such that $\underset{x\to\pm\infty}\lim b(x)=b_0.$ 4. $\underset{x\to\pm\infty}\lim b^\prime(x)=\underset{x\to\pm\infty}\lim b^{\prime\prime}(x)=0 .$ 5. $\int_\mathbb{R}x^2(|b(x)-b_0|+|b^{\prime}(x)|)\,dx<\infty$. The conditions 1-5 imply that the spectrum of the operator $\mathcal{A}$ consists of the interval $[0,\infty)$ and, possibly, several negative single eigenvalues. Denote by $H_{a}\subset L_2(\mathbb{R})$ an absolutely continuous subspace of the operator $\mathcal{A}$, and by $P_{a}$ the orthogonal projection into $L_2(\mathbb{R})$ on $H_{a}$. Let $\mathcal{A}_0=\mathcal{A}P_{a}$ denote the restriction of the operator $\mathcal{A}$ on $H_{a}$. For each $\lambda$ satisfying the condition $\mathrm{Re}\,\lambda\leqslant 0$, we define a random operator $\mathcal{R}_\lambda^t$ by setting $$\mathcal{R}_\lambda^tf(x)=\int_0^t e^{\lambda\tau}(P_{a}f)(\xi_x(\tau))e^{-\int_0^\tau V(\xi_x(s))\,ds} \,d\tau. $$ \textbf{Theorem 1.} 1. With probability 1 the operator $\mathcal{R}_\lambda^t$ is a bounded integral operator in $L_2(\mathbb{R})$ of the form $$\mathcal{R}_\lambda^tf(x)=\int_\mathbb{R}r_\lambda(t,x,y)f(y)\,dy,$$ where the last equality is also valid for $t=\infty$ in the case $\mathrm{Re }\,\lambda<0$. 2. For any $\lambda,t,x$ the function $r_\lambda(t,x,\cdot)\in W_2^\alpha$ for every $\alpha\in[0,\frac{1}{2})$. \textbf{Theorem 2.} 1. If $\mathrm{Re}\,\lambda< 0$ then for all $f\in H_{a}$ \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. If $\mathrm{Re}\,\lambda= 0$ and $\lambda\neq 0$ then for all $f\in H_{a}$ \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} For $\lambda=0$ the last equality (\ref{eq70}) holds for every $f\in \mathcal{D}(\mathcal{A}_0-\lambda I)^{-1}$. The work is supported by RSF, project \textnumero 22-21-00016.

Kovalevskii Artem
Video, Slides

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 the 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 the 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.

Zaporozhets Dmitry
Video

Random Polynomials Having No Real Zeros

In their 2002 paper, Dembo, Poonen, Shao and Zeitouni obtained a power asymptotic $n$ decreasing probability that a random polynomial of even degree $n$ with i.i.d. coefficients has no real zeros. The exact power exponent was not found, but it was conjectured that it is equal to $-3/4$. Only in the summer of 2021 FitzGerald, Tribe, and Zaboronski posted the work in the arXiv with its proof. In this talk we will consider a similar problem for random polynomials whose coefficients have binomial variance. These polynomials were first considered by Kostlan, Shub, and Smale in their works in the early 90s of the last century.

12:45 — 13:15

Tarasenko Anton, Lotov Vladimir
Video, Slides

Inequalities for the Characteristics of The Cusum Procedure in a Change Point Problem

We obtain an upper bound for the average delay time with a response to the presence of a change point and for the average time to a false alarm when a change point is detected using the CUSUM procedure.

Topchii Valentin
Video, Slides

Critical Galton-Watson Branching Processes with a Countable Set of Particle Types and Random Graphs

We consider genealogical trees of Galton-Watson branching processes and study the critical case corresponding to a one-vertex random tree with an independent, identically distributed number of edges for all vertices. The average number of edges coming out of a lower level vertex is $1$. One of the fundamental theorems for these processes is a Yaglom-type theorem, which states that processes that do not degenerate to a distant time $n$ contain at a given time the number of particles equal to this time $n$ times exponentially distributed random variable. It is convenient to describe these conditional processes in terms of reduced trees, which are obtained from genealogical trees by eliminating subtrees that do not reach level $n$. A more complex model of Galton-Watson branching processes with a countable set of particle types, in which the types of descendants are obtained by summing the parent type with independent identically distributed multidimensional random variables, can be represented as trees with weights of edges and vertices defined above in the one-dimensional case. We describe the averages and variances of a number of characteristics of reduced weighted trees, including the total weight of all vertices at a fixed level. We prove a number of limit theorems for reduced trees. The work was carried out within the framework of the state task of the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project FWNF-2022-0003.

Prokopenko Evgeny
Video, Slides

Multi-Normex approach for evaluating the sum of heavy tailed random vectors

We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'Normex' approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named $d$-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles. This is joint work with Marie Kratz.

13:15 — 13:30

Coffee-break

13:30 — 14:10

Korshunov Dmitry
Video

Large Deviations for Asymptotically Space Homogeneous Markov Chains in Two Dimensions

We discuss Markov chains in the 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
Video, Slides

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. 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
Video

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 $\{b_n\}$ such that ${S_n}/{b_n}$ converges in distribution towards the standard normal law as $n\to\infty$. Let $ T:=\inf\{k\geq1:S_k\leq g_k\} $ be the first crossing time 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 $\mathbf{P}(T>n)$. 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
Video

Prasolov Timofei

Moments of the First Descending Epoch for a Random Walk with Negative Drift

Efremov Egor

Moderate Deviations Principle for M-Dependent Random Variables in Sublinear Expectation Space

Ishkov Roman

Research of the Infection Model in Queues $M / M / k / 0$

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 yhe Regeneration Cycle for a Random Walk with Drift

Shelepova Anastasiya

Asymptotics of the Distribution of the Exit Time Beyond a Non-Increasing Boundary for a Compound Renewal Process

Lukyanov Andrey

Double Bootstrap Method for Tail Index Estimation by Expectiles

Trushin Alexander

Mathematics 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
Slides

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 Sakhanenko, 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).

Chebunin Mikhail
Video, Slides

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. This is joint work with Sergey Foss.

16:45 — 17:15

Borovkov Konstantin
Slides

Parisian Ruin with Random Deficit-Dependent Delays for Spectrally Negative L\'{e}vy 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\'{e}vy 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. This is joint work with Duy Phat Nguyen.

Logachev Artem, Mogulskii Anatolii
Video, Slides

Moderate Deviations Principles for Trajectories of Inhomogeneous Random Walks

We consider a normalized piecewise linear curve constructed from sums of independent random variables that may have different distributions. Under various moment conditions on random variables we present theorems containing the principles of moderate large deviations for such piecewise linear curves in the space of continuous functions on the interval [0,1]. We also point out the connection between the zone in which the principle of moderately large deviations is fulfilled and the moment that exists for random variables.

17:15 — 17:30

Welcome party + стенды

Coffee-break

17:30 — 18:10

Wachtel Vitali
Video, Slides

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.

Rybko Alexander
Video

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 to $1$ 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 a trivial invariant state when all the $N$ points merge 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
Video, Slides

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.

Zuyev Sergei
Video

Probing Harmony with Algebra (at least, statistically)

In a recent statistical study, US researchers 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.