UNIFORM CLT FOR MARKOV-CHAINS AND ITS INVARIANCE-PRINCIPLE - A MARTINGALE APPROACH

The convergence of stochastic processes indexed by parameters which ar e elements of a metric space is investigated in the context of an inva riance principle of the uniform central limit theorem (UCLT) for stati onary Markov chains. We assume the integrability condition on metric e ntropy with bracketing. An eventual uniform equicontinuity result is d eveloped which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequenc e as in Gordin and Lifsic.((7)) Then we use the chaining argument with stratification adapted from that of Ossiander.((11)) The results of t his paper generalize those of Levental((10)) and Ossiander.((11))