On uniform closeness of local times of Markov chains and i.i.d. sequences

Abstract

In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of n i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of [11].

An Improved Decoupling Inequality for Random Interlacements

“…At this point observe that, differently of what happens in [4], where we have two processes with the same size n ∈ N, here we have two processes (I u and M u ) with random cardinalities (N and N , respectively) which have different laws. Hence, in a first step we will couple this random cardinalities (see (7)), and then we will use a "resampling" technique (like in [4]) to complete the construction of the coupling between I u and M u (see Figure 2).…”

“…Therefore, using Proposition 5.1 of [4] (with δ 0 from this proposition equal to 1), for k, ∈ N, ≥ 4, we obtain, on the sets D ∩ C k, i ∩ {N 1 = k} ∩ {N 2,2 = } such that C k, i ⊂ G k, ,…”

“…Thus, I u K will be the trace left on K by the elements of I u . Next we present a method to construct I u , in the same spirit as [4].…”

“…In this section we present the construction of a coupling between I u and M u , using the soft local times technique. This coupling is inspired from the one described in [4], Section 4.…”

Conditional decoupling of random interlacements

“…However, the polynomial error term in (1.3) can complicate one's life in many applications (and, e.g. in the case when the diameters of these sets are of the same order as the distance between them, (1.3) is simply of no use); on the other hand, while (1.3) can be improved to some degree [2], the error term there should always be at least polynomial, as (1.2) shows. To circumvent this difficulty, one first may note that usually the "interesting" events/functions are monotone (i.e., increasing or decreasing).…”