The strong consistency of maximum likelihood estimates for nonlinear systems

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 converge almost surely to the correct parameter value. This is established by use of a submartingale property of the sequence of maximized likelihood ratios together with a technique first employed by Wald [3] in the case of independent identically distributed random variables.