Suppose I have two stochastic matrices $A_1$ and $A_2$. It is well known that the convergence rate of the Markovian chain generated by a stochastic matrix is determined by the absolute value of its second largest eigenvalue.
Let us denote the value for $A_i$ ($i=1,2$) as $\lambda_i $.
Now, consider the following Markovian chain, in which $A_1$ and $A_2$ alternate. That is, starting from a vector $x$, the series is $\{x, A_1 x, A_2 A_1 x, A_1 A_2 A_1 x, A_2 A_1 A_2 A_1 x, \ldots \}$
The question is, how is the convergence rate of this chain?
Could it be faster than the purely $A_1$ chain and the purely $A_2$ chain? Or it must be inbetween?