A multivariate extension of the Erdos-Taylor theorem
成果类型：
Article
署名作者：
Lygkonis, Dimitris; Zygouras, Nikos
署名单位：
University of Warwick
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-024-01267-3
发表日期：
2024
页码：
179-227
关键词：
摘要：
The Erdos-Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if L(N )is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then pi/logN L(N )converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if L-N((1-2)) = Sigma(N)(n=1) 1({Sn(1)=Sn(2)}), then pi/logN L-N((1, 2)) converges in distribution to an exponential random variable of parameter one. We prove that for every h >= 3, the family {pi/logN L-N((i, j))}(1 <= i < j <= h), of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdos-Taylor theorem. We also discuss connections to directed polymers in random environments.
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