Limit Theorems for Affine Markov Walks Conditioned to Stay Positive
Résumé
Consider the real Markov walk $S_n = X_1+ \dots+ X_n$ with increments $\left(X_n\right)_{n\geqslant 1}$ 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 $\left( y+S_n \right)_{n\geqslant 1}$
from the positive part of the real line.
We investigate the asymptotic behaviour of the probability of the event $\tau_y \geqslant n$
and of the conditional law of $y+S_n$ given $\tau_y \geqslant n$ as $n \to +\infty$.