A law of the iterated logarithm for stochastic integrals

• Published in: Stochastic processes and their applications Vol. 47; no. 2; pp. 215 - 228
• Main Author: Wang, Jia-gang
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.1993; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: estimation of spectral function; Exact sciences and technology; law of the iterated logarithm; law of the iterated logarithm semimartingale square integrable martingale stochastic calculus estimation of spectral function; Limit theorems; Mathematics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; semimartingale; square integrable martingale; stochastic calculus; 60G44; stochastic calculus; law of the iterated logarithm; square integrable martingale; 62M15; semimartingale; 60H05; estimation of spectral function; 60F15; Spectral function; Stochastic calculus; Statistical estimation; Law of iterated logarithm; Quadratic form; Semimartingale; Stochastic integral
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/0304-4149(93)90015-V

By using the Itô calculus, a law of the iterated logarithm (LIL) is established for stochastic integrals with respect to locally square integrable martingales. Let M = ( M t , t ⩾ 0) be a d-dimensional locally square integrable martingale, B = ( B t ) be a d-dimensional predictable process and X = B T · M. If lim t→∞ 〈 X〉 t = ∞ a.s., | B t | 2 = o(〈 X〉 β t ), β < 1 and there exists a majorant measure for the Lévy system of M, then P lim sup t→∞ X t 2〈 X〉 t log log(〈 X〉 tV e =1 =1 . As an application of this LIL, a LIL for the quadratic form of i.i.d. random variables is given. In particular, let { ξ n , n ⩾ 1} be a sequence of i.i.d. random variables with Eξ n = 0, Eξ 2 n = σ 2 > 0 and Eξ 4 n = μ 4 < ∞ . If F n(λ)= 1 2πn ∫ −λ λ ∑ j=1 n χ n e − inμ 2 dμ . Then for all λ ϵ (0, π], P lim sup n→∞ n (F n(λ)−λσ 2⧸π) 2 log logn = 2σ 4πλ+(μ 4− 3σ 4)λ 2 π =1 .