A Representation for Non-colliding Random Walks
نویسنده
چکیده
Let D0(R+ ) denote the space of cadlag paths f : R+ → R with f(0) = 0. For f, g ∈ D0(R+ ), define f ⊗ g ∈ D0(R+ ) and f g ∈ D0(R+ ) by (f ⊗ g)(t) = inf 0≤s≤t [f(s) + g(t) − g(s)], and (f g)(t) = sup 0≤s≤t [f(s) + g(t) − g(s)]. Unless otherwise deleniated by parentheses, the default order of operations is from left to right; for example, when we write f ⊗ g ⊗ h, we mean (f ⊗ g) ⊗ h. Define a sequence of mappings Γk : D0(R+ ) → D0(R+ ) by Γ2(f, g) = (f ⊗ g, g f), and, for k > 2, Γk(f1, . . . , fk) = (f1 ⊗ f2 ⊗ · · · ⊗ fk, Γk−1(f2 f1, f3 (f1 ⊗ f2), . . . , fk (f1 ⊗ · · · ⊗ fk−1))). Let N1, . . . , Nn be the counting functions of independent Poisson processes on R+ with respective intensities μ1 < μ2 < · · · < μn. Our main result is that the conditional law of N1, . . . , Nn, given N1(t) ≤ · · · ≤ Nn(t), for all t ≥ 0, is the same as the unconditional law of Γn(N). From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in
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Non - Colliding Random Walks , Tandem Queues , and Discrete Orthogonal Polynomial Ensembles
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