Invariance principle for the random walk Sample Clauses

Invariance principle for the random walk. It remains to show that the invariance principle in Section 3.2.2 for the chain (w(n))n∈N0 implies the invariance principle of Theorem 3.2. To this aim, consider the random process ( ˜ ) (L) Z − cτ ˜ τ (L) k Sn(k) with k∈N Sn(k) = √n . (3.40) ˜ ˜ We first construct a coupling that allows us to compare Sn with Υn. After that we pass from Sn to St defined in (3.6). Fix w ∈ W and ϵ ∈ (0, 1). Consider an enlarged probability space, with law Pє,w, on which there exist a sequence (rk)k∈N distributed according to P¯µ,(0,0)(r ∈ · | H1), with r as in (3.10), and a sequence (w(k))k∈N distributed according to Pw. On this enlarged probability space, by using (3.33), we can couple (rk)k∈N and (w(k))k∈N in a recursive manner such that Pє,w(ri+1 = w1(i + 1) | r1, . . . , ri, w1(1), . . . , w1(i)) ≥ 1 − Ψ(kL) (3.41) ∑ on the event {rl = w1(l), i − k + 1 ≤ l ≤ i} for any k ∈ {1, . . . , i}. Hence, by (3.41) and the fact that k∈N Ψ(kL) < ∞, we have a sequence k0(ϵ) < ∞, with k0(ϵ) → ∞ as ϵ → 0, such that Pє,w(∃k ≥ k0(ϵ): rk /= w1(k)) ≤ ϵ. (3.42) Next, recall Lemma 2.6, fix T > 0, and let IT = 2(T + 1)/(Jr−L) with J = lim inf E¯ µ,(0,0) (T (L)). (3.43) 1 From (3.42), we have that ( Pє,w sup k0(є)≤k≤nIT S˜n(k) − S˜n(k0(ϵ)) − Υn(k/n) − Υn(k0(ϵ)/n) 1 > 0) ≤ ϵ. (3.44) Moreover, for any δ > 0, lim P¯µ,(0,0)  sup  1 t≤τ (L) Zt 1 > δ√n lim  ≤ P¯µ,(0,0) (τ (L) > δ√n) = 0, (3.45) and, by using (2.49) with α > 1, we get P¯µ,(0,0)  sup  sup { Zt − Z (L) > 3δ√n | H1 1≤ k≤n τ (L)≤t≤τ (L) k  µ,(0,0) µ,(0,0) ( sup {τ (L) − τ (L)} > δ√n) = 1 − [1 − P¯ (τ (L) > δ√n)]n