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If \(T\) is the transition matrix of a regular Markov process (so for some \(k\) all the entries of \(T^{k}\) are positive), we know there is a probability vector \(P_{\infty}\) so that if \(P_{0}\) is any initial probability vector, then \(\lim _{k \rightarrow \infty} T^{k} P_{0}=P_{\infty}\).
Show that the matrix \(\lim _{k \rightarrow \infty} T^{k}\) has all of its columns equal to \(P_{\infty}\).
in Mathematics by Platinum (129,882 points) | 86 views

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