Note on minimum contrast estimates forMarkov processes

Summary In the present paper it is shown that the concept of minimum contrast estimates (m.c.e.) considered inPfanzagl 1969a] for independent and identically distributed (iid) observations can be modified to cover stationary discrete timeMarkov processes admitting a unique stationary distribution which dominates the transition probabilities (Condition (S)). Sufficient conditions on the measurability and strong consistency of m.c.e. stated inPfanzagl 1969a] for the idd case are reformulated to give sufficient conditions for the existence of measurable m.c.e. and their strong consistency for such processes (section 1). The proofs of the main theorems are only sketched, because they are nearly the same as those given inPfanzagl 1969a] for the iid case.The concept of m.c.e. covers maximum likelihood estimates (m.l.e.) as a special case; therefore an application of the results to m.l.e. yields sufficient conditions for the existence of measurable m.l.e. and their strong consistency if the parameter space is compact metrizable or locally compact with countable base (Section 2). These conditions are weaker than the usual regularity conditions (see for exampleBillingsley 1961b] and the references cited there) and under the assumption that the transition probabilities as well as the stationary distribution are absolutely continuous with respect to a -finite measure they can be expressed in terms of the corresponding equivalence classes of transition densities. This seems to be more transparent than the conditions given byRoussas 1965]. In Section 3 asymptotic normality of m.c.e. is proved under conditions which correspond to those used inRoussas 1968] for the case of m.l.e.