Some random environment population processes are considered. Using a coupling argument, a criticality condition is obtained. The criticality parameter governs whether the process dies out or fluctuates about a positive finite constant, which is explicitly computed.

A random walk on the line is observed at times n1, n1 + n2,.... If the steps have 2k - 2 finite moments then the kth moment can be estimated consistently if and only if [Sigma]n2-ki = [infinity].