Yuri Yakubovich

Saint Petersburg State University | SPBU · Department of Theory of Probability and Mathematical Statistics

We consider a sequence of i.i.d. random variables, (ξ)=(ξi)i=0,1,2,⋯, Eξ0=0, Eξ02=1, and subordinate it by a doubly stochastic Poisson process Π(λt), where λ≥0 is a random variable and Π is a standard Poisson process. The subordinated continuous time process ψ(t)=ξΠ(λt) is known as the PSI-process. Elements of the triplet (Π,λ,(ξ)) are supposed to...

Weconsider a PSI-process, that is a sequence of random variables (&), i = 0.1,…, which is subordinated by a continuous-time non-decreasing integer-valued process N(t): <K0 = Ç N (ty Our main example is when /V(t) itself is obtained as a subordination of the standard Poisson process 77(s) by a non-decreasing Lévy process S(t): N(t) = 77(S(t)).The va...

We present a relatively simple and mostly elementary proof of the Lévy–Khintchine formula for subordinators. The main idea is to study the Poisson process time-changed by the subordinator. This is a compound Poisson process which is easy to investigate using elementary probabilistic techniques. It turns out that its rate equals the value of the Lap...

Построена стохастическая модель информационного канала со случайной интенсивностью и случайной нагрузкой. Мы исследуем модель информационного потока, где происходит проекция части информации «прошлого» на «настоящее», а часть информации исчезает. Момент «настоящего» дополняется инновациями, которые представляются в виде замещений исчезнувшей информ...

We present a relatively simple and mostly elementary proof of the L\'evy--Khintchine formula for subordinators. The main idea is to study the Poisson process time-changed by the subordinator. The technical tools used are conditional expectations, probability generating function and convergence of discrete random variables.

The class of minimal difference partitions\(\text {MDP}(q)\) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least \(q\ge 0\). In a recent series of papers by A. Comtet and collaborators, the \(\text {MDP}(q)\) ensemble with uniform measure was interpreted as a combinatorial model...

The class of minimal difference partitions MDP(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q\ge 0. In a recent series of papers by A. Comtet and collaborators, the MDP(q) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with f...

This article presents a limit theorem for the gaps $\widehat{G}_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of a sample of size $n$ from a random discrete distribution on the positive integers $(P_1, P_2, \ldots)$ governed by a residual allocation model (also called a Bernoulli sieve) $P_j:= H_j \prod_...

The lattice of the set partitions of $[n]$ ordered by refinement is studied. Given a map $\phi: [n] \rightarrow [n]$, by taking preimages of elements we construct a partition of $[n]$. Suppose $t$ partitions $p_1,p_2,\dots,p_t$ are chosen independently according to the uniform measure on the set of mappings $[n]\rightarrow [n]$. The probability tha...

We consider shifts $\Pi_{n,m}$ of a partially exchangeable random partition $\Pi_\infty$ of $\mathbb{N}$ obtained by restricting $\Pi_\infty$ to $\{n+1,n+2,\dots, n+m\}$ and then subtracting $n$ from each element to get a partition of $[m]:= \{1, \ldots, m \}$. We show that for each fixed $m$ the distribution of $\Pi_{n,m}$ converges to the distrib...

We describe the distribution of frequencies ordered by sample values in a random sample of size $n$ from the two parameter GEM$(\alpha,\theta)$ random discrete distribution on the positive integers. These frequencies are a $($size$-\alpha)$-biased random permutation of the sample frequencies in either ranked order, or in the order of appearance of...

We show that in a sample of size $n$ from a GEM$(0,\theta)$ random discrete distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of the sample, with the convention $G_{n:n} := X_{1:n} - 1$, are distributed like the first $n$ terms of an infinite sequence of independent geometric$(i/(i+...

We show that the maximal value in a size $n$ sample from GEM$(\theta)$ distribution is distributed as a sum of independent geometric random variables. This implies that the maximal value grows as $\theta\log(n)$ as $n\to\infty$. For the two-parametric GEM$(\alpha,\theta)$ distribution we show that the maximal value grows as a random factor of $n^{\...

We show that the expected value of the descent after the first maximum in a sample of i.i.d. discrete random variables, as the sample size grows, behaves asymptotically up to vanishing terms as the expectation of the maximal value minus the expectation of a sampled random variable, provided the latter is finite. We also show that the expected value...

For a subfamily of multiplicative measures on integer partitions we give conditions for properly rescaled associated Young diagrams to converge in probability to a certain deterministic curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples.

We examine the total number of collisions Cn in the Λ-coalescent process which starts with n particles. A linear growth and a stable limit law for Cn are shown under the assumption of a power-like behaviour of the measure Λ near 0 with exponent 0 < α < 1.

We examine the total number of collisions $C_n$ in the $\Lambda$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure $\Lambda$ near 0 with exponent $0<\alpha<1$.

We consider the occupancy problem where balls are thrown independently at infinitely many boxes with fixed positive frequencies. It is well known that the random number of boxes occupied by the first n balls is asymptotically normal if its variance Vn tends to infinity. In this work, we mainly focus on the opposite case where Vn is bounded, and der...

We investigate the limiting distribution of the fluctuations of the maximal summand in a random partition of a large integer with respect to a multiplicative statistics. We show that for a big family of Gibbs measures on partitions (so-called generalized Bose–Einstein statistics) this distribution is the well-known Gumbel distribution which usually...

A class of random discrete distributions $P$ is introduced by means of a recursive splitting of unity. Assuming supercritical branching, we show that for partitions induced by sampling from such $P$ a power growth of the number of blocks is typical. Some known and some new partition structures appear when $P$ is induced by a Dirichlet splitting.

We consider the slicing of Young diagrams into slices associated with summands that have equal multiplicities. It is shown that for the uniform measure on all partitions of an integer n, as well as for the uniform measure on partitions of an integer n into m summands with m ∼ Anα, α ≤ 1/2, all slices after rescaling concentrate around their limit s...

Given two positive integers m n, we consider the set of partitions = (1 ;:::; '; 0 ;::: ), 1 2 ::: ,o fn such that the sum of its parts over a xed increasing subsequence (aj )i sm: a1 + a2 + = m. We show that the number of such partitions does not depend on n if m is either constant and small relatively to n or depend on n but is close to its large...

We study the limiting behavior of uniform measures on finite-dimensional simplices as the dimension tends to infinity and a discrete analog of this problem, the limiting behavior of uniform measures on compositions. It is shown that the coordinate distribution of a typical point in a simplex, as well as the distribution of summands in a typical com...

We consider the set of all partitions of a number n into distinct summands (the so-called strict partitions) with the uniform distribution on it and study fluctuations of a random partition near its limit shape, for large n. The use of geometrical language allows us to state the problem in terms of the limit behavior of random step functions (Young...

We consider the uniform distribution on the set of partitions of integer n with cn numbers of summands, c>0 is a positive constant. We calculate the limit shape of such partitions, assuming c is constant and n tends to infinity. If c→∞ then the limit shape tends to known limit shape for unrestricted number of summands. If the growth is slower than...

We consider a set of partitions of natural number n on distinct summands with uniform distribution. We investigate the limit shape of the typical partition as n → ∞, which was found in [A. M. Vershik, Funct. Anal. Appl, 30 (1996), pp. 90-105], and fluctuations of partitions near its limit shape. The geometrical language we use allows us to reformul...

A family of multiplicative statistics on integer partitions that are symmetric under the conjunction of Young diagrams is constructed. It is shown that all symmetric multiplicative statistics lie in this family. A limit shape of Young diagrams under these statistics is found.

In this paper we deal with some class of continuous partition lattices, introduced by A. Bjorner in 1987 as a limit of finite partition lattices. We describe the direct way how to construct the continuous partition lattice in terms of measurable partitions of the measure space and present a simple realization of such lattices. Secondly, we give a r...

This paper contains two results on the asymptotic behavior of uniform probability measure on partitions of a finite set as its cardinality tends to infinity. The first one states that there exists a normalization of the corresponding Young diagrams such that the induced measure has a weak limit. This limit is shown to be a δ-measure supported by th...