A Note on the Consistency of Maximum Likelihood Estimates for Finite Families of Stochastic Processes

We consider families of stochastic processes indexed by a finite number of alternative parameter values. For general classes of stochastic processes it is shown that maximum likelihood estimates convergence almost surely to the correct parameter value. This established by use of a submartingale property of the sequence of maximized likelihood ratios together with a technique first employed by Wald [24] in the case of independently identically distributed random variables.