Asymptotically stable random walks of index 1<α<2 killed on a finite set

Abstract

running head: random walks killed on a finite set key words: one dimensional random walk; first passage time; killed at the origin; in a domain of attraction, transition probability, stable process AMS Subject classification (2010): Primary 60G50, Secondary 60J45. AbstractFor a random walk on the integer lattice Z that is attracted to a strictly stable process with index α ∈ (1, 2) we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in a natural range of the space and time variables. The situation is relatively simple when the limit stable process has jumps in both positive and negative directions; in the other case when the jumps are one sided rather interesting matters are involved and detailed analyses are necessitated.

Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set

“…that follow immediately from Lemma 6.2 (combined with (3.19)) and Lemma 6.5(ii) of [15], respectively. As for −2λ n ≤ y < 0, in view of the identity p n {0} (x, y) = p n {0} (−y, −x) what has been just proved yields the bound p n {0} (x, y) < C/n for x > 2λ n .…”

“…(the reversed inequality with C −1 M replaced by C M also holds), where C M is a positive constant (see Proposition 2.3(ii) of [15] as well as the comment given right after it). We also have…”

“…Corresponding lemmas for the case σ 2 = ∞ Here we suppose that (2.11) is satisfied and prove Lemmas 3.3 and 3.4 below that correspond to Lemmas 3.1 and 3.2, respectively, and will be used for the proofs of Theorems 3 and 4. We shall use the following large deviation estimate: [4] [15] in which the transition function of the killed walk denoted by Q n A (x, y) is defined slightly differently from but agrees with p n A (x, y) whenever y / ∈ A.…”

“…The proof of the lemma proceeds parallel to that of Lemma 3.2 and we point out only main steps after stating the results from [15] needed for it. Put for x > 0 D n (x) = a(x)λ n n 2 + x nλ n .…”

“…As for P5 also missing in the above, we need a corresponding one in a dual form, which will be presented below in (4.24). It is shown in [15,Theorem 6] that as N → ∞…”