On the prospective minimum of the random walk conditioned to stay non-negative

TL;DR

This paper investigates the asymptotic behavior of the minimum of a conditioned random walk within a stable law domain of attraction, revealing five possible limiting distributions based on parameter relationships.

Contribution

It characterizes the prospective minimum of a non-negative conditioned random walk in the stable law domain, identifying five different limiting behaviors depending on parameter regimes.

Findings

Five distinct limiting expressions for the conditioned minimum.

Dependence of limit forms on relationships between parameters r, k, and n.

Asymptotic distribution results for the minimum of the walk under conditioning.

Abstract

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 $n$.