Year: 2024
Type: article
Abstract: Let S 0 = 0 , S n = X 1 + … + X n , n ≥ 1 , $$\begin{array}{} \displaystyle S_{0}=0,\quad S_{n}=X_{1}+\ldots+X_{n},\, n\geq 1, \end{array}$$ be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants an that provide convergence as n → ∞ of the distributions of the sequence {Sn /an , n = 1, 2, …} to this stable law. Let L r,n = min r≤m≤n Sm be the minimum of the random walk on the interval [r, n]. It is shown that lim r , k , n → ∞ P L r , n ≤ y a k | S n ≤ t a k , L 0 , n ≥ 0 , t ∈ 0 , ∞ , $$\begin{array}{} \displaystyle \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\
Authors: Vladimir A. Vatutin, Elena E. Dyakonova
Institution: Steklov Mathematical Institute
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Russian Science Foundation
Award: G4823391070
OpenAlex ID: W4414251111