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För andra betydelser, se Ångermanland (olika betydelser). .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .in...

4 I want to determine the generating function of the numbers of visits up to a specific time. Formally: let ${X_{n}}_{n geq 0}$ be a markov chain on the state space $S = {1,ldots,r}$ with transition matrix $P = (p_{i,j})_{1 leq i,j leq r}$ and initial probability vector $alpha = (alpha_{i} : i in S)$ with $alpha_{i} = mathbb{P}(X_{0} = i)$ . Let $q(n_{1},ldots,n_{r})$ be the probability that up to time $n-1$ the state $1$ is visited $n_{1}$ -times, $ldots$ ,state $r$ is visited $n_{r}$ -times and $F_{n}(x_{1},ldots,x_{r}) = sum_{n_{1}+ cdots +n_{r} = n}q(n_{1},ldots,n_{r})x_{1}^{n_{1}}cdots x_{r}^{n_{r}}$ be the corresponding generating function of $q(n_{1},ldots,n_{r})$ . Now I want to show that $$ F_{n}(x_{1},ldots,x_{r}) = (alpha_{1}x_{1},ldots,alpha_{r}x_{r}) left( P D_{r}right)^{n-1} e, $$ where $e = (...