  Consider a Markov chain $(X_n)_{n\geqslant 0}$ with values in the state space
$\mathbb X$. Let $f$ be a real function on $\mathbb X$ and set $S_0=0,$ $S_n =
f(X_1)+\cdots + f(X_n),$ $n\geqslant 1$. Let $\mathbb P_x$ be the probability
measure generated by the Markov chain starting at $X_0=x$. For a starting point
$y \in \mathbb R$ denote by $\tau_y$ the first moment when the Markov walk
$(y+S_n)_{n\geqslant 1}$ becomes non-positive. Under the condition that $S_n$
has zero drift, we find the asymptotics of the probability $\mathbb P_x (
\tau_y >n )$ and of the conditional law $\mathbb P_x ( y+S_n\leqslant
\cdot\sqrt{n} | \tau_y >n )$ as $n\to +\infty.$

Probability