Limit theorems for affine Markov walks conditioned to stay positive

Consider the real Markov walk $S_n=X_1+\cdots +X_n$ with increments ${\left(X_n\right)}_{n\ge 1}$ 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 ${(y+S_n)}_{n\ge 1}$ from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event ${\tau }_y\ge n$ and of the conditional law of $y+S_n$ given ${\tau }_y\ge n$ as $n\to +\mathrm{\infty }$.