The problem

Let us suppose $P$ and $P+Q$ are two stochastic matrices (non-negative coefficients with $P \mathbf{1} = \mathbf{1}$ and $(P+Q) \mathbf{1} = \mathbf{1}$).

Let $n \geq 1$, I am seeking for a bound on $\|(P+Q)^n - P^n\|_F$, if possible depending on $\|Q\|_F$

Discussion

When $P$ and $P+Q$ are irreductible aperiodic stochastic matrices, I was thinking of studying separately :

• $\|(P+Q)^n - \underset{n \to \infty}{\lim} (P+Q)^n\|_F$
• $\|P^n - \underset{n \to \infty}{\lim}P^n\|_F$
• $\|\underset{n \to \infty}{\lim} (P+Q)^n - \underset{n \to \infty}{\lim}P^n\|_F$

Do you know if there is a straightforward way to bound these quantities ?

In general, can we use the convergence in average of the ergodic theorem to find a similar decomposition ?