Vladimir Vatutin

Vladimir Vatutin
Russian Academy of Sciences | RAS · Steklov Mathematical Institute

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256
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2,606
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November 1977 - present
Russian Academy of Sciences
Position
  • Leading researcher

Publications

Publications (256)
Article
Пусть $Z_{n},n=0,1,2,…\} $ - критический ветвящийся процесс в случайной среде, а $\{ S_{n},n=0,1,2,…\} $ - его сопровождающее случайное блуждание. Известно, что если приращения этого случайного блуждания принадлежат (без центрирования) области притяжения устойчивого закона, то существует такая правильно меняющаяся на бесконечности последовательност...
Article
Рассматривается критический ветвящийся процесс $\{Y_n, n\geq 0\}$, эволюционирующий в случайной среде, порожденной последовательностью независимых одинаково распределенных случайных величин, к каждому поколению которого присоединяется ровно один иммигрант. Обозначим через $\mathcal A_i(n)$ событие, состоящее в том, что все индивидуумы, существующие...
Article
We consider a two-type decomposable branching process (Z1(n),Z2(n)), n≥0, where type 1 particles may produce particles of types 1 and 2 while type 2 particles can give birth only to type 2 particles. Let Zi(m,n),0≤m<n, i = 1, 2 be the number of type i particles existing in the process at moment m < n and having a positive number of descendants at m...
Article
A survey of results in the theory of multitype branching processes evolving in a random environment is presented. Bibliography: 104 titles.
Article
We study properties of a p-type subcritical branching process in random environment initiated at moment zero by a vector z = (z1, .., zp) of particles of different types. For p = 1 the class of processes we consider corresponds to the so-called strongly subcritical case. It is shown that the survival probability of this process up to moment n behav...
Preprint
Full-text available
We consider a critical branching process $Y_{n}$ in an i.i.d. random environment, in which one immigrant arrives at each generation. Let $% \mathcal{A}_{i}(n)$ be the event that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the conditional distribution of $Y_{n}$ given $\mathcal{A...
Article
Full-text available
We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event Ai(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{...
Article
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We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty...
Article
A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvec...
Article
Рассматривается неразложимый ветвящийся процесс Гальтона-Ватсона со счетным множеством типов частиц. Предполагая, что процесс является критическим, а частицы некоторых (или всех) его типов могут иметь бесконечную дисперсию числа непосредственных потомков, мы описываем асимптотическое поведение вероятности невырождения процесса и доказываем условную...
Article
We consider an indecomposable Galton-Watson branching process with a countable set of types. Assuming that the process is critical and may have infinite variance of the offspring sizes of some (or all) types of particles we describe the asymptotic behaviour of the survival probability of the process and establish a Yaglom-type conditional limit the...
Article
В работе дан обзор результатов по теории многотипных ветвящихся процессов, эволюционирующих в случайной среде. Библиография: 104 названия.
Article
Пусть $\mathcal{F}_{n,N}=\{ F_{n,N}\} $ - множество всех лесов, каждый из которых образован $N$ помеченными корневыми деревьями $T_{1},T_{2},\ldots ,T_{N}$, имеющими в совокупности $n+N$ вершин (считая корни), причем вершинам приписаны несовпадающие метки (числа от $1$ до $n+N$). Обозначим $\lambda (u)$ метку, приписанную вершине $u$ леса $F_{n,N}$...
Article
Рассматривается докритический ветвящийся процесс в случайной среде, компоненты которой одинаково распределены и независимы. Предполагается, что к каждому поколению частиц присоединяется ровно один иммигрант. Пусть $\mathcal{A}_i(n)$ - событие, состоящее в том, что все частицы основного процесса, живущие в момент $n$, являются потомками иммигранта,...
Preprint
Full-text available
We consider a subcritical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We consider the event $% \mathcal{A}_{i}(n)$ that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$ and investigate the asymptotic probability of this extreme event wh...
Preprint
We study properties of a $p-$type subcritical branching process in random environment initiated at moment zero by a vector $\mathbf{z}=\left( z_{1},..,z_{p}\right) $\ of particles of different types. Assuming that the process belongs to the class of the so-called strongly subcritical processes we show that its survival probability to moment $n$\ be...
Article
Full-text available
We consider a population of particles with unit lifetime. Dying, each particle produces offspring whose size depends on the random environment specifying the reproduction law of all particles of the given generation and on the number of relatives of the particle. We study the asymptotic behavior of the survival probability of the population up to a...
Article
A critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is ini...
Preprint
Full-text available
We consider an indecomposable Galton-Watson branching process with countably infinitely many types. Assuming that the process is critical and allowing for infinite variance of the offspring sizes of some (or all) types of particles we describe the asymptotic behavior of the survival probability of the process and establish a Yaglom-type conditional...
Article
Исследуется асимптотическое поведение вероятности невырождения многотипных ветвящихся процессов в случайной среде. В случае одного типа частиц класс рассматриваемых процессов соответствует промежуточно докритическим процессам. При достаточно общих предположениях о форме производящих функций законов размножения частиц доказано, что вероятность невыр...
Article
The asymptotic behavior of the survival probability for multi-type branching processes in a random environment is studied. In the case where all particles are of one type, the class of processes under consideration corresponds to intermediately subcritical processes. Under fairly general assumptions on the form of the generating functions of the la...
Article
Исследуются свойства докритического ветвящегося процесса в случайной среде с $p$ типами частиц, начальное число частиц различных типов в котором задается вектором $\mathbf{z}=(z_{1},\ldots,z_{p})$. В случае $p=1$ класс рассматриваемых нами процессов соответствует строго докритическим ветвящимся процессам в случайной среде с одним типом частиц. Дока...
Preprint
Full-text available
We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event $\mathcal{A}_i(n)$ that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the asymptotic probability of this event when $n$ is la...
Preprint
Full-text available
We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow \rho ,\ n\rightarrow \infty $, which is a standard condition in f...
Preprint
Full-text available
We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods. We prove that, the tail distribution decays at an exponential rate or more quickly, according to the solution of an equation. The main tools are the change of measure and some lim...
Preprint
Full-text available
A critical branching process with immigration which evolve in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the aboriginal population we investigate the tail distribution of the so-called life period of the process, i.e., the length of the time interval between the moment when the proc...
Preprint
Full-text available
We study the asymptotic behaviour of the survival probability of a multi-type branching processes in random environment. The class of processes we consider corresponds, in the one-dimensional situation, to the intermediately subcritical case. We show under rather general assumptions on the form of the offspring generating functions of particles tha...
Article
Let {Z(n),n≥ 0} be a critical Galton–Watson branching process with finite variance for the offspring number of particles. Assuming that 0 <Z(n) ≤ ϕ(n), where either ϕ(n) =an for some a>0orϕ(n) =o(n) asn →∞, we study the structure of the process {Z(m, n), 0 ≤ m ≤ n}, where Z(m, n) is the number of particles in the initial process at moment m ≤ n hav...
Article
Рассматривается многотипный ветвящийся процесс Гальтона-Ватсона в случайной среде, задаваемой последовательностью независимых одинаково распределенных случайных величин. В предположении, что приращение $X$ сопровождающего случайного блуждания, порожденного логарифмами перроновых корней матриц средних этого процесса, удовлетворяет условиям $\mathbf{...
Article
Доказана условная предельная теорема о распределении размера популяции на начальном этапе эволюции слабо докритического ветвящегося процесса в случайной среде при условии его невырождения к далекому моменту времени.
Article
Рассматривается $b$-арное плоское корневое дерево $T,$ вершинам которого независимо и равновероятно присвоены цвета, обозначаемые буквами алфавита $\mathcal{A}=\{ A_{1}<A_{2}<...<A_{m}\} .$ Вершина $u\in T$ является предком вершины $v\in T$ ($u\prec v)$, если путь, ведущий по ребрам от корня дерева к вершине $v,$ проходит через вершину $u$. Обознач...
Preprint
Full-text available
We consider a population of particles with unit life length. Dying each particle produces offspring whose size depends on the random environment specifying the reproduction law of all particles of the given generation and on the number of relatives of the particle. We study the asymptotic behavior of the survival probability of the population up to...
Article
We study the asymptotic behavior of the survival probability of a multi-type branching process in a random environment. In the one-dimensional situation, the class of processes considered corresponds to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process give...
Article
A critical Bellman-Harris branching process { Z ( t ), t ≥ 0} with finite variance of the offspring number is considered. Assuming that 0 < Z ( t ) ≤ φ ( t ), where either φ ( t ) = o ( t ) as t → ∞ or φ ( t ) = at , a >0, we study the structure of the process where Z ( s , t ) is the number of particles in the initial process at moment s which eit...
Preprint
Full-text available
Let $\left\{ Z(t), t\geq 0\right\} $ be a critical Bellman-Harris branching process with finite variance for the offspring size of particles. Assuming that $0<Z(t)\leq \varphi (t)$, where either $\varphi (t)=o(t)$ as $t\rightarrow \infty $ or $\varphi (t)=at,\, a>0$, we study the structure of the process $% \left\{ Z(s,t),0\leq s\leq t\right\} ,$ w...
Article
A two-type critical decomposable branching process with discrete time is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. Assuming that the offspring distributions of particles of both types may...
Article
Full-text available
Let $\left\{ Z(n),n\geq 1\right\} $ be a critical Galton-Watson branching process with finite variance for the offspring size of particles. Assuming that $0<Z(n)\leq \varphi (n)$, where either $\varphi (n)=an$ for some $a>0$ or $\varphi (n)=o(n)$ as $n\rightarrow \infty $, we study the structure of the process $% \left\{ Z(m,n),0\leq m\leq n\right\...
Article
Доказана предельная теорема о распределении числа частиц в близких к критическому ветвящихся процессах с финальным типом частиц при условии невырождения таких процессов к далекому моменту времени.
Article
Full-text available
We study the asymptotic behaviour of the survival probability of a multitype branching process in random environment. The class of processes we consider here corresponds, in the one-dimensional situation, to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process...
Chapter
In this chapter, the intermediate subcritical case is investigated under the annealed approach. This is located on the borderline between weakly and strongly subcritical BPREs. The passage corresponds to a phase transition in the model, thus a particularly rich behavior can be expected from the intermediate case. This is reflected in the results pr...
Chapter
The study of BPREs essentially uses properties of the ordinary random walk. This chapter contains some basic results needed in the sequel. It introduces some sequences of random variables related to given assumption. These random variables are called ladder heights. Ladder epochs and ladder heights will repeatedly appear in the subsequent arguments...
Chapter
In this chapter, the properties of critical BPREs are investigated under the quenched approach. The quenched approach gives more possibilities in the analogous situation. Contrary to the annealed approach, the transformation of measures is performed from P to P+, and the product of measures P × P and transform it to P- × P+ is also used. The chapte...
Article
Full-text available
We investigate the limit behavior of supercritical multitype branching processes in random environments with linear fractional offspring distributions and show that there exists a phase transition in the behavior of local probabilites of the process affected by strongly and intermediately supercritical regimes. Some conditional limit theorems can a...
Article
Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a...
Article
Full-text available
Using the annealed approach we investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in i.i.d. random environment. We show under rather general assumptions on the form of the offspring generating functions of particles that the probability of survival up to generation $n$ of the process i...
Chapter
This paper is a survey of some recent results on the asymptotics of the survival probability, limit theorems conditioned on survival or attaining a high level of single-type subcritical branching processes in independent and identically distributed random environments.
Article
Full-text available
A critical branching process $\left\{Z_{k},k=0,1,2,...\right\} $ in a random environment generated by a sequence of independent and identically distributed random reproduction laws is considered.\ Let $Z_{p,n}$ be the number of particles at time $p\leq n$ having a positive offspring number at time $n$. \ A theorem is proved describing the limiting...
Article
A critical branching process $\left\{ Z_{k},k=0,1,2,...\right\} $ in a random environment is considered. A conditional functional limit theorem for the properly scaled process $\left\{ \log Z_{pu},0\leq u<\infty \right\} $ is established under the assumptions $Z_{n}>0$ and $p\ll n$. It is shown that the limiting process is a Levy process conditione...
Article
A decomposable strongly critical Galton–Watson branching process, with N types of particles labeled 1, 2,…, N, is considered in which particles of type i may produce offspring of types j ≧ i only. Functional limit theorems are proved describing the structure of the reduced process generated by the initial one, and the distributions of the birth mom...
Article
Full-text available
The asymptotic behavior, as $n\rightarrow \infty $ of the conditional distribution of the number of particles in a decomposable critical branching process $\mathbf{Z}% (m)=(Z_{1}(m),...,Z_{N}(m)),$ with $N$ types of particles at moment $m=n-k,\, k=o(n),$ is investigated given that the extinction moment of the process is $n$.
Article
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We consider a Galton-Watson process $\mathbf{Z}% (n)=(Z_{1}(n),Z_{2}(n))$ with two types of particles. Particles of type 2 may produce offspring of both types while particles of type 1 may produce particles of their own type only. Let $Z_{i}(m,n)$ be the number of particles of type $i$ at time $m<n$ having offspring at time $n$. Assuming that the p...
Article
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The asymptotic behavior, as $n\rightarrow \infty $ of the probability of the event that a decomposable critical branching process $\mathbf{Z}(m)=(Z_{1}(m),...,Z_{N}(m)),$ $m=0,1,2,...,$ with $N$ types of particles dies at moment $n$ is investigated and conditional limit theorems are proved describing the distribution of the number of particles in t...
Article
A decomposable Galton–Watson branching process with N types of particles labeled 1, 2,…,N is considered in which a type i parent may produce individuals of types j ≥ i only. This model may be viewed as a stochastic model of the development of a population whose individuals may be located at one of the N islands, the location of a particle being con...
Article
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Let {S-n, n >= 0} g with S-0 = 0 be a random walk with negative drift and let tau(x) = min {k > 0 : S-k < -x}, x >= 0 : Assuming that the distribution of the i.i.d. increments of the random walk is absolutely continuous with subexponential density we describe the asymptotic behavior, as n -> infinity, of the probabilities P (tau(x) = n) and P (S-n...
Article
An asymptotic behavior of increments of the renewal functions generated by the distributions with tails varying at ±∞ regularly with index β∈(0,0·5] is investigated.
Article
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We study the asymptotics of the survival probability for the critical and decomposable branching processes in random environment and prove Yaglom type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in constant environment
Article
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A decomposable strongly critical Galton-Watson branching process with $N$ types of particles labelled $1,2,...,N$ is considered in which a type~$i$ parent may produce individuals of types $j\geq i$ only. This model may be viewed as a stochastic model for the sizes of a geographically structured population occupying $N$ islands, the location of a pa...
Article
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We consider random walks with finite second moment which drifts to $-\infty$ and have heavy tail. We focus on the events when the minimum and the final value of this walk belong to some compact set. We first specify the associated probability. Then, conditionally on such an event, we finely describe the trajectory of the random walk. It yields a de...
Article
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We investigate a two-type critical Bellman--Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order $o(t^{-2})$; the tail of the life-length distribution of the second type particles is regularly varying at infinity with index $-\beta$, $\beta \in (0,1]$; at time $t=0$...
Article
A critical indecomposable two-type Bellman-Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter β ∈ (0, 1]. It is shown that, contrary to the critical indecomposable B...
Article
This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of...
Article
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Let $Z_{n}$ be the number of individuals in a subcritical BPRE evolving in the environment generated by iid probability distributions. Let $X$ be the logarithm of the expected offspring size per individual given the environment. Assuming that the density of $X$ has the form $$p_{X}(x)=x^{-\beta -1}l_{0}(x)e^{-\rho x}$$ for some $\beta >2,$ a slowly...
Article
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The Bernoulli sieve is a random allocation scheme obtained by placing independent points with the uniform [0,1] law into the intervals made up by successive positions of a multiplicative random walk with factors taking values in the interval (0,1). Assuming that the number of points is equal to n we investigate the weak convergence, as n tends to i...
Article
A time-continuous branching random walk on the lattice ℤd , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced...
Article
A two-type pure decomposable branching process in a random environment is considered. Each particle of this process may produce offspring of its own type only. Let exp{Xk(i)} be the mean number of children produced by a particle of type i = 1, 2 of generation k. Assuming that Xk(2) = -Xk(1) with probability 1 and that the random walk Sn(1) = X1(1)+...
Article
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A population has two types of individuals, each occupying an island. One of those, where individuals of type 1 live, offers a variable environment. Type 2 individuals dwell on the other island, in a constant environment. Only one-way migration (1->2) is possible. We study the asymptotics of the survival probability in critical and subcritical cases...
Article
Evolutionary branching is analysed in a stochastic, individual-based population model under mutation and selection. In such models, the common assumption is that individual reproduction and life career are characterised by values of a trait, and also by population sizes, and that mutations lead to small changes ϵ in trait value. Then, traditionally...
Article
For a critical branching process evolving in a randomenvironment and having geometric distributions of offspring sizes, we study the tail behavior of the distributions of the total size of the population and the maximal number of particles in a generation up to the moment of extinction of the process.
Article
Using methods of multitype branching processes in random environment counted by random characteristics, we study the tail distribution of busy periods and some other characteristics of the branching-type polling systems in which the service disciplines, input parameters, and service time distributions are changing in a random manner.
Article
Details are given about the Bernoulli Society for Mathematical Statistics and Probability, including the Society's history, organization, member benefits, and past and present activities. Reports are given on the International Conference “Modern Stochastics: Theory and Applications II” held September 7–11, 2010 in Kyiv, Ukraine; the International C...
Article
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This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in IID random environments. The key assumptions of the family of population models in question are: non-overlapping generations, independent reproduction of particles...
Article
A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment $n$ given survival to this moment....
Article
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For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within...
Article
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First a population model with one single type of individuals is considered. Individuals reproduce asexually by splitting into two, with a population-size-dependent probability. Population extinction, growth and persistence are studied. Subsequently the results are extended to such a population with two competing morphs and are applied to a simple m...
Article
Acknowledgment of the 80th birthday of Yuri Vasilievich Prokhorov, outstanding contributor to the development of modern probability theory.
Article
We establish convergence to the Kingman coalescent for the genealogy of a geographically-or otherwise-structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it...