  For a Markov chain $Y$ with values in a Polish space, consider the entrance
chain obtained by sampling $Y$ at the moments when it enters a fixed set $A$
from its complement $A^c$. Similarly, consider the exit chain, obtained by
sampling $Y$ at the exit times from $A^c$ to $A$. We use the method of inducing
from ergodic theory to study invariant measures of these two types of Markov
chains in the case when the initial chain $Y$ has a known invariant measure. We
give explicit formulas for invariant measures of the entrance and exit chains
under certain recurrence-type assumptions on $A$ and $A^c$, which apply even
for transient chains. Then we study uniqueness and ergodicity of these
invariant measures assuming that $Y$ is topologically recurrent, topologically
irreducible, and weak Feller.
  We give applications to random walks in $R^d$, which we regard as
"stationary" Markov chains started under the Lebesgue measure. We are mostly
interested in dimension one, where we study the Markov chain of overshoots
above the zero level of a random walk that oscillates between $-\infty$ and
$+\infty$. We show that this chain is ergodic, and use this result to prove a
central limit theorem for the number of level crossings for random walks with
zero mean and finite variance of increments.
