Self-adjusting stochastic processes

Let At(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that At+1(h, k) [greater-or-equal, slanted] At(h, k). If the self-adjustment (due to transition h --> kx) is At + 1(h, j) = [lambda]At(h, j) + (1 - [lambda])[delta]jk (0 < [lambda] < 1), then with probability 1 the process is eventually periodic. If A0(i, j) < 1 for all i, j and if the self-adjustment satisfies At + 1(h, k) = [phi](At(h, k)) with [phi](x) twice differentiable and increasing, x < [phi](x) < 1 for 0 [less-than-or-equals, slant] x < 1,[phi](1) = [phi]'(1) = 1, then, with probability 1, lim At does not exist.