The Annals of Probability

Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks

Xia Chen

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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 In=#{(k1,…,kp)∈[1,n]p;S1(k1)=⋯=Sp(kp)} 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.

Article information

Source
Ann. Probab., Volume 32, Number 4 (2004), 3248-3300.

Dates
First available in Project Euclid: 8 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1107883353

Digital Object Identifier
doi:10.1214/009117904000000513

Mathematical Reviews number (MathSciNet)
MR2094445

Zentralblatt MATH identifier
1067.60071

Subjects
Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F10: Large deviations 60F15: Strong theorems 60G50: Sums of independent random variables; random walks 60J55: Local time and additive functionals 60J65: Brownian motion [See also 58J65]

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

Citation

Chen, Xia. Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab. 32 (2004), no. 4, 3248--3300. doi:10.1214/009117904000000513. https://projecteuclid.org/euclid.aop/1107883353


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