Limit theorems for affine Markov walks conditioned to stay positive

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 $n \to +\infty$.