Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix

Hello,

Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero.

As a part of my research, I am interested in upper-bounding the following expression: $\parallel\log_\star \lim_{p\rightarrow\infty}A^p\parallel_2$, in terms of $\parallel A\parallel_2$.

Does anyone have an idea?

Thank you.

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For $A$ sufficiently nice, its invariant distribution $p$ is given by $p = 1^*(I − A + 11^∗)^{−1}$, where here $1$ denotes a vector of ones. Moreover, $A^\infty = 1p$. Unfortunately, this is of precisely a form that will break if you try to use the Sherman-Morrison formula. – Steve Huntsman Nov 9 '12 at 23:02
Isn't it entirely possible that some entry of $A^p$ can be zero, in which case the lhs becomes unbounded, while the 2-norm of $A$ is well defined? – Suvrit Nov 9 '12 at 23:32
For example, consider the identity matrix! – Suvrit Nov 9 '12 at 23:33
@Suvrit: I would bet that $A$ is supposed to correspond to an irreducible Markov chain. – Steve Huntsman Nov 10 '12 at 1:39
Sorry, it is indeed an irreducible Markov Chain. I edited. – Daniel86 Nov 10 '12 at 21:12