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October 2004 Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks
Xia Chen
Ann. Probab. 32(4): 3248-3300 (October 2004). DOI: 10.1214/009117904000000513

Abstract

Let α([0,1]p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d−2)<d and d≥2, we prove $$\lim_{t\to\infty}t^{-1}\log \mathbb{P}\bigl\{\alpha([0,1]^{p})\ge t^{(d(p-1))/2}\bigr\}=-\gamma_{\alpha}(d,p)$$ with the right-hand side being identified in terms of the the best constant of the Gagliardo–Nirenberg inequality. Within the scale of moderate deviations, we also establish the precise tail asymptotics for the intersection local time $$I_n=\#\{(k_1,…,k_p)∈[1,n]^p ; S_1(k_1)=⋯=S_p(k_p)\}$$ run by the independent, symmetric, ℤd-valued random walks S1(n), …,Sp(n). Our results apply to the law of the iterated logarithm. Our approach is based on Feynman–Kac type large deviation, time exponentiation, moment computation and some technologies along the lines of probability in Banach space. As an interesting coproduct, we obtain the inequality $$\bigl({\mathbb{E}}I_{n_{1}+\cdots +n_{a}}^{m}\bigr)^{1/p}\le \sum_{\mathop{k_{1}+\cdots +k_{a}=m}\limits_{k_{1},\ldots,k_{a}\ge 0}}{\frac{m!}{k_{1}!\cdots k_{a}!}}\bigl({\mathbb{E}}I_{n_{1}}^{k_{1}}\bigr)^{1/p}\cdots \bigl({\mathbb{E}}I_{n_{a}}^{k_{a}}\bigr)^{1/p}$$ in the case of random walks.

Citation

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Xia Chen. "Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks." Ann. Probab. 32 (4) 3248 - 3300, October 2004. https://doi.org/10.1214/009117904000000513

Information

Published: October 2004
First available in Project Euclid: 8 February 2005

zbMATH: 1067.60071
MathSciNet: MR2094445
Digital Object Identifier: 10.1214/009117904000000513

Subjects:
Primary: 60B12 , 60F10 , 60F15 , 60G50 , 60J55 , 60J65

Keywords: Gagliardo–Nirenberg inequality , Intersection local time , large (moderate) deviations , Law of the iterated logarithm

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 4 • October 2004
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