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On the prospective minimum of the random walk conditioned to stay nonnegative.

On the prospective minimum of the random walk conditioned to stay nonnegative.

Authors :
Vatutin, Vladimir A.
Dyakonova, Elena E.
Source :
Discrete Mathematics & Applications. Dec2024, Vol. 34 Issue 6, p337-362. 26p.
Publication Year :
2024

Abstract

Let S 0 = 0 , S n = X 1 + ... + X n , n ≥ 1 , 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 Lr,n = minr≤m≤nSm 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 , ∞ , can have five different expressions, the forms of which depend on the relationships between the parameters r, k and n. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
09249265
Volume :
34
Issue :
6
Database :
Academic Search Index
Journal :
Discrete Mathematics & Applications
Publication Type :
Academic Journal
Accession number :
181547307
Full Text :
https://doi.org/10.1515/dma-2024-0030