= Empirical occupation proportions of a finite Markov chain
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{title2=$\widehat\pi_j=n^{-1}\sum_{t=0}^{n-1}\mathbf1_{\{X_t=j\}}$}
The empirical occupation proportion is the fraction of the observed <Markov chain> states equal to $j$. These nonnegative fractions sum to one. To simulate them from a <transition matrix>, sample an initial state and then draw each successive state from the current state's row, accumulating visit counts. They are empirical frequencies for a finite path, whether or not the chain is already stationary. Long-run convergence to a stationary distribution requires the corresponding recurrence and irreducibility hypotheses.
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