Abstract
We show that the function $h(x)=\prod_{i \lt j}(x_j-x_i)$ is harmonic for any random walk in $R^k$ with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber $W=\{x\colon x_1 \lt x_2 \lt \cdots \lt x_k\}$ onto a point where $h$ vanishes, we define the corresponding Doob $h$-transform. For certain special cases, we show that the marginal distribution of the conditioned process at a fixed time is given by a familiar discrete orthogonal polynomial ensemble. These include the Krawtchouk and Charlier ensembles, where the underlying walks are binomial and Poisson, respectively. We refer to the corresponding conditioned processes in these cases as the Krawtchouk and Charlier processes. In [O'Connell and Yor (2001b)], a representation was obtained for the Charlier process by considering a sequence of $M/M/1$ queues in tandem. We present the analogue of this representation theorem for the Krawtchouk process, by considering a sequence of discrete-time $M/M/1$ queues in tandem. We also present related results for random walks on the circle, and relate a system of non-colliding walks in this case to the discrete analogue of the circular unitary ensemble (CUE).
Highlights
We are concerned with probability distributions on Zk of the formP(x) = 1 h(x)[2] P(x), Z x ∈ Zk, (1.1)where P is some well-known distribution on Zk, Z is the normalizing constant, and the function h is given by k h(x) =. (1.2) i,j=1 i
Where P is some well-known distribution on Zk, Z is the normalizing constant, and the function h is given by k h(x) =
First we show that the function h is harmonic for any random walk in Rk with exchangeable increments, provided the required moments exist. (Note that h is defined on Rk .) Define the Weyl Chamber as
Summary
See [Jo01, Jo00a, Jo00b], and references given there, for models which lead to these ensembles, connections between these ensembles, and their asymptotic analysis as k → ∞ These ensembles are discrete analogues of ensembles which arise as eigenvalue distributions in random matrix theory (see [Me91], for example) and marginal distributions for non-colliding diffusion processes [Bi94, Dy62, Gr99, HW96, KO01]. The invariant distribution for the system of non-colliding walks on the circle is a discrete orthogonal polynomial ensemble which can be regarded as the discrete analogue of the circular unitary ensemble (CUE) The latter arises in random matrix theory as the law of the eigenvalues of a random unitary matrix chosen according to Haar measure on the group of unitary matrices of a particular dimension.
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