Conditions equivalent to consistency of approximate MLEʹs for stochastic processes

Maximum likelihood and approximate maximum likelihood estimates of parameters of random processes are considered. A mild regularity is assumed under which these estimates exist. Conditions necessary and sufficient for consistency of all approximate maximum likelihood estimates are established. These conditions are first applied to i.i.d. observations and the result is shown to be in some sense sharper than what is known from the literature. Then new consistency results are obtained by applying these conditions to observations from various concrete classes of discrete and continuous processes.