Abstract
The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.(7) Then we use the chaining argument with stratification adapted from that of Ossiander.(11) The results of this paper generalize those of Levental(10) and Ossiander.(11)
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Bae, J., Levental, S. Uniform CLT for Markov chains and its invariance principle: A martingale approach. J Theor Probab 8, 549–570 (1995). https://doi.org/10.1007/BF02218044
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DOI: https://doi.org/10.1007/BF02218044