On a discrete approximation of a skew stable L\'{e}vy process

Iksanov and Pilipenko (2023) defined a skew stable L\'{e}vy process as a scaling limit of a sequence of perturbed at $0$ symmetric stable L\'{e}vy processes (continuous-time processes). Here, we provide a simpler construction of the skew stable L\'{e}vy process as a scaling limit of a sequence of perturbed at $0$ standard random walks (random sequences).Comment: 23 pages, submitted to a journa

Some functionals for random walks and critical branching processes in extreme random environment

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 $\alpha$-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 $\infty$ to a random variable having an $\alpha$-stable distribution. Let $S_{0}=0,$% \begin{equation*} L_{n}:=\min \left( S_{1},...,S_{n}\right) ,\tau _{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 $$ exists we prove several limit theorems describing the asymptotic behavior of the functionals \begin{equation*} \mathbf{E}\left[ e^{S_{\tau _{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 environmentComment: 28 page

Random walks conditioned to stay non-negative and branching processes in non-favorable random environment

Let $\{S_n,n\geq 0\}$ be a random walk whose increments belong without centering to the domain of attraction of an $\alpha$-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 $S_n\leq \varphi (n)=o(a_n),$ 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.Comment: 35 pages, 27 reference

Random Sieves and Generalized Leader-election Procedures

A random sieve of the set of positive integers N is an infinite sequence of nested subsets N = S0 ⊃ S1 ⊃ S2 ⊃ · · · such that Sk is obtained from Sk−1 by removing elements of Sk−1 with the indices outside Rk and enumerating the remaining elements in the increasing order. Here R1 , R2 , . . . is a sequence of independent copies of an infinite random set R ⊂ N. We prove general limit theorems for Sn and related functionals, as n → ∞

Critical branching processes in a sparse random environment

We introduce a branching process in a sparse random environment as an intermediate model between a Galton–Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event