The potential function and ladder heights of a recurrent random walk on $\mathbb {Z}$ with infinite variance

Estimates of Potential functions of random walks on $Z$ with zero mean and infinite variance and their applications

“…tends to zero uniformly for 0 < x < R, which is the same as the equivalence relation of (5.22). The second relation of the proposition is Lemma 6.4 of [28].…”

“…2 < ∞ and a(−x) converges to a constant that is positive if the walk is not right-continuous (cf. [28,Theorem 2]). Hence we may consider only the case α ≤ β ≤ 2α − 1.…”

Recurrent random walks on $\mathbb{Z}$ with infinite variance: transition probabilities of them killed on a finite set

“…This is valid at least for all −n 1/α < y < 0 and can be extended to y ≤ −n 1/α . For the proof of the extension we have only to observe that if y ≤ −n 1/α , then the RHS is not less than c/n 1/α if z < −n 1/α with some c > 0 while for z ≤ −n 1/α , (5.17) follows from Lemma 5.2 (note that [21] or (6.19)), and from (5.17) we deduce…”

The two-sided exit problem for a random walk on $\mathbb{Z}$ and having infinite variance II

“…which take less simple forms for E x [a(S T )] and g Ω (x, y). Here (3.1) follows from Corollary 1 of [26] and (3.2) from the identity g {0} (x, y) = g(x, y)…”