= Markov chain return bridge
{c}
{title2=$\mathbb P_i(M_n=j\mid M_N=i)=\dfrac{(P^n)_{ij}(P^{N-n})_{ji}}{(P^N)_{ii}}$}
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.
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