arXiv subjects
Publications and source records attributed to Vladimir Vatutin.
We study the evolution of the population size distribution of a critical Galton-Watson process with infinite variance of the offspring size of particles assuming that the population size is unusually small at the distant moment $n$ of observation.
Let $\{Z_{m},m\geq 0\}$ be a critical branching process in random environment and $\{S_{m},m\geq 0\}$ be its associated random walk. Assuming that the increments distribution of the associated random walk belongs without centering to the domain of attraction of an $α$-stable law we prove conditional limit theorems describing, as $n\rightarrow \infty $, the distribution the number of particles in the process $\{Z_{m},0\leq m\leq n\}$ given $Z_{n}>0$ and $S_{n}\leq const$.
We study the asymptotic behavior of small deviation probabilities for the critical Galton-Watson processes with infinite variance of the offspring sizes of particles and apply the obtained result to investigate the structure of a reduced critical Galton-Watson process.
Let \begin{equation*} S_{0}=0,\quad S_{n}=X_{1}+...+X_{n},\ n\geq 1, \end{equation*} be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants $a_{n}$, that provide convergence as $n\rightarrow \infty $ of the distributions of the elements of the sequence $\left\{ S_{n}/a_{n},n=1,2,...\right\} $ to this stable law. Let $L_{r,n}=\min_{r\leq m\leq n}S_{m}$ be the minimum of the random walk on the interval $[r,n]$. It is shown that \begin{equation*} \lim_{r,k,n\rightarrow \infty }\mathbf{P}\left( L_{r,n}\leq ya_{k}|S_{n}\leq ta_{k},L_{0,n}\geq 0\right) ,\, t\in \left( 0,\infty \right), \end{equation*} can have five different expressions, the forms of which depend on the relationships between the parameters $r,k$ and $n$.
Let $\left\{ S_{n},n\geq 0\right\} $ be a random walk whose increment distribution belongs without centering to the domain of attraction of an $% α$-stable law, i.e., there are some scaling constants $a_{n}$ such that the sequence $S_{n}/a_{n},n=1,2,...,$ weakly converges, as $% n\rightarrow \infty $ to a random variable having an $α$-stable distribution. Let $S_{0}=0,$% \begin{equation*} L_{n}:=\min \left( S_{1},...,S_{n}\right) ,τ_{n}:=\min \left\{ 0\leq k\leq n:S_{k}=\min (0,L_{n})\right\} . \end{equation*}% Assuming that $S_{n}\leq h(n),$ where $h(n)$ is $o(a_{n})$ and $% \lim_{n\rightarrow \infty }h(n)\in \lbrack -\infty ,+\infty ]$ exists we prove several limit theorems describing the asymptotic behavior of the functionals \begin{equation*} \mathbf{E}\left[ e^{S_{τ_{n}}};S_{n}\leq h(n)\right] \end{equation*}% as $n\rightarrow \infty $. The obtained results are applied for studying the survival probability of a critical branching process evolving in an extremely unfavorable random environment. Key words: random walk, branching processes, random environment, survival probability, unfavorable environment
Let $\{S_n,n\geq 0\} $ be a random walk whose increments belong without centering to the domain of attraction of an $α$-stable law $\{Y_t,t\geq 0\}$, i.e. $S_{nt}/a_n\Rightarrow Y_t,t\geq 0,$ for some scaling constants $a_n$. Assuming that $S_0=o(a_{n})$ and $S_n\leq φ(n)=o(a_n),$ we prove several conditional limit theorems for the distribution of $S_{n-m}$ given $m=o(n)$ and $\min_{0\leq k\leq n}S_k\geq 0$. These theorems complement the statements established by F. Caravenna and L. Chaumont in 2013. The obtained results are applied for studying the population size of a critical branching process evolving in non-favorable environment.
Let $\left\{ Z_{n},n=0,1,2,...\right\} $ be a critical branching process in random environment and let $\left\{ S_{n},n=0,1,2,...\right\} $ be its associated random walk. It is known that if the increments of this random walk belong (without centering) to the domain of attraction of a stable law, then there exists a sequence $a_{1},a_{2},...,$ slowly varying at infinity such that the conditional distributions \begin{equation*} \mathbf{P}\left( \frac{S_{n}}{a_{n}}\leq x\Big|Z_{n}>0\right) ,\quad x\in (-\infty ,+\infty ), \end{equation*}% weakly converges, as $n\rightarrow \infty $ to the distribution of a strictly positive and proper random variable. In this paper we supplement this result with a description of the asymptotic behavior of the probability \begin{equation*} \mathbf{P}\left( S_{n}\leq φ(n);Z_{n}>0\right) , \end{equation*}% if $φ(n)\rightarrow \infty $ \ as $n\rightarrow \infty $ in such a way that $φ(n)=o(a_{n})$.
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 $n$\ behaves for large $n$\ as $C(\mathbf{z})λ^{n}$\ where $λ$\ is the upper Lyapunov exponent for the product of mean matrices of the process and $C(\mathbf{z})$% \ is an explicitly given constant. We also demonstrate that the limiting conditional distribution of the number of particles given the survival of the process for a long time does not depend on the vector $\mathbf{z}$ of the number of particles initiated the process.
We consider 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 again. We prove that the tail distribution decays with exponential rate. The main tools are the change of measure and some conditional limit theorems for random walks.
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 process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.
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 that the probability of survival up to generation $n$ of the process initiated at moment zero by a single particle of any type is of order $λ^{n}n^{-1/2}$ for large $n,$ where $λ\in (0,1)$ is a constant specified by the Lyapunov exponent of the mean matrices of the process.
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 $0 0$, we study the structure of the process $% \left\{ Z(s,t),0\leq s\leq t\right\} ,$ where $Z(s,t)$ is the number of particles in the process at moment $s$ in the initial process which either survive up to moment $t$ or have a positive offspring number at this moment.
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 $0 0$ or $φ(n)=o(n)$ as $n\rightarrow \infty $, we study the structure of the process $% \left\{ Z(m,n),0\leq m\leq n\right\} ,$ where $Z(m,n)$ is the number of particles in the process at moment $m\leq n$ having a positive number of descendants at moment $n$.
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 given its survival for a long time.
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 also be obtained from the representation of generating functions.
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 conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.
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 the process $\mathbf{Z}(\cdot)$ at moment $m<n,$ given that the extinction moment of the process is $n$. These limit theorems may be considered as the statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees having a fixed hight.
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