 Markov chain return bridge

For a time-homogeneous Markov chain started at $i$, condition on a return to $i$ at a fixed later time $N$, assuming that event has positive probability. The Markov property factors the probability of a visit to $j$ at time $n$ and a later return into the two transition probabilities in the numerator. Conditional probability then gives the formula. At the midpoint with $j=i$ and $N=2n$, it reduces to $p_n^2/p_{2n}$, where $p_n=(P^n{)}_{ii}$. If $p_n\sim a{\lambda }^n$ for $a,\lambda >0$, this midpoint return probability tends to $a$ even when unconditional returns tend to zero.