On infinite products of stochastic matrices (Q1189630)

For a given sequence \(\{Q_ 0,Q_ 1,\ldots\}\) of \(n\times n\) stochastic matrices let \(\{T_ 0,T_ 1,\ldots\}\) be a sequence of products taken in a certain order of multiplication. One says that weak ergodicity obtains for this order of multiplication if \((T_ N)_{ik}-(T_ N)_{jk}\to 0\) as \(N\to\infty\) for every \(i,j\) and \(k\). One says that strong ergodicity obtains for this order of multiplication if weak ergodicity obtains and \((T_ N)_{ij}\) converges to a limit as \(N\to\infty\) for every \(i\) and \(j\). Conditions are established for weak ergodicity of products taken in an arbitrary order, and strong ergodicity of the backward products \(M_ N=Q_ NQ_{N-1}\ldots Q_ 1Q_ 0\). Strong ergodicity is studied for products taken in an arbitrary order.