Limit theorems for affine Markov walks conditioned to stay positive

TL;DR

This paper studies the long-term behavior of affine Markov walks conditioned to stay positive, focusing on exit times and the distribution of the process given it remains positive, providing new limit theorems in this context.

Contribution

It establishes new limit theorems for affine Markov walks conditioned to stay positive, extending understanding of their asymptotic behavior and exit time probabilities.

Findings

Asymptotic estimates for the probability $ au_y extgreater= n$

Limit laws for the process conditioned on staying positive

Characterization of the distribution of $y+S_n$ given $ au_y extgreater= n

Abstract

Consider the real Markov walk $S_n = X_1+ \dots+ X_n$ with increments $\left(X_n\right)_{n\geq 1}$ defined by a stochastic recursion starting at $X_0=x$. For a starting point $y>0$ denote by $\tau_y$ the exit time of the process $\left( y+S_n \right)_{n\geq 1}$ from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event $\tau_y \geq n$ and of the conditional law of $y+S_n$ given $\tau_y \geq n$ as $n \to +\infty$.