On almost sure behavior of stable subordinators over rapidly increasing sequences

Abstract

Let (X(t), t ≥ (0) with X(0) = 0 be a stable subordinator with index 0 < α < 1 and let (tk) be an increasing sequence such that tk+1/tk → ∞ as k → ∞. Let (at) be a positive nondecreasing function of t such that a(t)/t 1. Define Y (t) = X(t + a(t)) − X(t) and Z(t) = X(t) − X(t − a(t)), t > 0. We obtain law-of-the-iterated-logarithm results for (X(tk)), (Y (tk)) and Z(tk), properly normalized.