ERGODICITY AND FUNCTIONAL CENTRAL LIMIT THEOREMS FOR A CLASS OF MARKOV PROCESSES WITH APPLICATIONS TO NONLINEAR AUTOREGRESSIVE MODELS (INVARIANT, PROBABILITY).

摘要

Let S be a nonempty set, (GAMMA) a set of mappings on S into S, and let P be a probability on (GAMMA). Let X(,n):n (GREATERTHEQ) 0 be a Markov process with state space S and transition probability p(x,B) = P( (gamma) (ELEM) (GAMMA):(gamma) (x) (ELEM) B ).;In the case that S is a complete separable metric space and (GAMMA) is the set of all contractions of S, we prove some sufficient conditions on P for the existence of a unique invariant probability (pi) and for the functional central limit theorem to hold for every Lipschitzian f (ELEM) L('2) ((pi)) on S under a mild additional assumption.;We also consider a Strassen-type law of the interated logarithm for X(,n) and applications to non-linear autoregressive models.;In case S = finite product of intervals, (GAMMA) is the set of all nondecreasing functions on S into S and P( (gamma) (ELEM) (GAMMA):(gamma) (x) (LESSTHEQ) x(,0) for all x (ELEM) S ) > 0, P( (gamma) (ELEM) (GAMMA):(gamma) (x) (GREATERTHEQ) x(,0) for all x (ELEM) S ) > 0 for some x(,0), the adjoint Markov operator is shown to be a uniformly strict contraction with respect to the Kolmogorov distance, which implies the existence of a unique invariant probability (pi) and the functional central limit theorem holds for every f (ELEM) L('2) ((pi)) with (INT)d(pi) = 0 which is of bounded variation on compact subsets of S.