2
$\begingroup$

Let $S$ be a finite set of states. Let $(M_n)$ be a sequence of transitions on $S$; that is, for every natural number $n$, $M_n$ is a non-negative $|S| \times |S|$ matrix whose rows sum up to 1. Suppose that the limit $M :=\lim_{n \to \infty} M_n$ exists (and is a transition matrix; here limit means that for every entry $(i, j)$, the sequence $(M_n(i,j))_n$ converges). I am interested in the Cesaro mean $\lim_{N \to\infty} \frac{1}{N}\sum_{n=1}^N \left(\prod_{i=1}^nM_i\right)$: under which conditions on $(M_n)$ does the Cesaro mean exist? Plainly, if $(M_n)$ converges quickly to $M$ (for example, $\sum_{n=1}^\infty \|M_n- M\| < \infty$) then the Cesaro mean exists. Another trivial case in which the Cesaro mean exists is when there is only one stable set under $M$. There is literature when the rank of the limit $M$ is 1. I am interested in the case where $M$ is general.

A more general question concerns the case when $(M_n)$ is a stochastic process: the transition at stage $n$ depends on past realizations (and maybe also on other random factors).

Eilon

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.