= Excursion occupation measure
{title2=$\gamma_j^k=\mathbb E_k\sum_{n=0}^{T_k-1}\mathbf1_{\{X_n=j\}}$}
For a <recurrent Markov chain>, start at $k$ and stop at its first positive return $T_k$. The expected visits to each state before that return form the excursion occupation measure. Its $k$th coordinate is one. Counting the return endpoint instead of the start gives the same measure. It is invariant by shifting the occupation sum one time step and using the <Markov property>. Any <Markov chain invariant measure> normalized by $x_k=1$ dominates it: iterate the stationarity equations on the state space with $k$ removed, then retain the nonnegative excursion-path terms.
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