Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks
| dc.creator | Chen, Xia | |
| dc.date | 2005-03-25 | |
| dc.date.accessioned | 2026-07-07T05:18:28Z | |
| dc.date.available | 2026-07-07T05:18:28Z | |
| dc.description | 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\ge 2, we prove lim_{t\to\infty}t^{-1}\log P\bigl{α([0,1]^p)\ge t^{(d(p-1))/2}\bigr}=-γ_α(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)\in [1,n]^p;S_1(k_1)=... =S_p(k_p)} run by the independent, symmetric, Z^d-valued random walks S_1(n),...,S_p(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(EI_{n_1+... +n_a}^m\bigr)^{1/p}\le \sum_{k_1+... +k_a=m\limits_{k_1,...,k_a\ge 0}}\frac{m!}{k_1!... k_a!}\bigl(EI_{n_1}^{k_1}\bigr)^{1/p}... \bigl(EI_{n_a}^{k_a}\bigr)^{1/p} in the case of random walks. | |
| dc.description | Published at http://dx.doi.org/10.1214/009117904000000513 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0503593 | |
| dc.identifier | http://arxiv.org/abs/math/0503593 | |
| dc.identifier | Annals of Probability 2004, Vol. 32, No. 4, 3248-3300 | |
| dc.identifier | doi:10.1214/009117904000000513 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/74674 | |
| dc.subject | Probability | |
| dc.subject | 60B12, 60F10, 60F15, 60G50, 60J55, 60J65. (Primary) | |
| dc.title | Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks | |
| dc.type | text |