A Representation for Non-Colliding Random Walks
Neil O'Connell, BRIMS, HP Labs
Marc Yor, Universite Pierre et Marie Curie
Abstract
We define a sequence of mappings $Gamma_k:D_0(R_+)^kto
D_0(R_+)^k$ and prove the following result: Let $N_1,ldots,N_n$ be the
counting functions of independent Poisson processes on $R_+$ with respective
intensities $mu_1<mu_2<cdots <mu_n$. The conditional law of
$N_1,ldots,N_n$, given that $$N_1(t)lecdotsle N_n(t), mbox{ for all
}tge 0,$$ is the same as the unconditional law of $Gamma_n(N)$. From
this, we deduce the corresponding results for independent Poisson processes
of equal rates and for independent Brownian motions (in both of these cases
the conditioning is in the sense of Doob). This extends a recent
observation, independently due to Baryshnikov (2001) and Gravner, Tracy
and Widom (2001), which relates the law of a certain functional of Brownian
motion to that of the largest eigenvalue of a GUE random matrix.
Our main result can also be regarded as a generalisation of Pitman's representation
for the 3-dimensional Bessel process.
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