A chain with complete connections Sample Clauses

A chain with complete connections. We construct a chain with complete connections that carries the necessary information relative to the evolution of the path of the RW Z in (2.29), together with the states of the RE ξ inside the truncated cones visited by the path between regeneration times. Lemma 3.3 below uses the n-cone-mixing property to control the dependence of the future evolution of the chain on its past. In particular, we will see that the influence of the past decays as fast as the correlations in the RE. We start by defining the relevant state space. Recall (3.1) and for N ∈ N let PN = x = (x(0), x(1), . . . , x(N )) ∈ CN (0)N : { } (3.7) be the set of possible paths of the process Z within the truncated cone CN (0), where x(i + 1) ∼ x(i) stands for |x1(i + 1) − x1(i)| = 1, x2(i + 1) − x2(i) = 1. Define ∪ T = {N} × PN × EN , (3.8) N∈N where EN = {ξt(z): (z, t) ∈ CN (0)} is the set of possible values of the environment ξ in the cone CN (0). Let W = {T ∪ {s}}N (3.9) be the set of infinite vectors with components either in T or equal to the stopping symbol s, with the restriction that if wk = s then wi = s for i ≥ k. Note that, for fixed k∈N L ∈ N, the sequence of regeneration times (τ (L)) in (2.35), together with the path Z, determine an infinite sequence r = (r1, r2, . . . ) ∈ W given by τk )}) rk = (S k+1,L ( (L) τk +j Sk+1,L ) {, ξ (z): (z, t) ∈ C Sk+1,L (Z (L) ∈ T , k ∈ N, with Sk,L = τ (L) − τ (L) − L, k ∈ N. (3.11) k k−1 1