Limit theorems for radial random walks on p × q-matrices as p tends to infinity

Abstract

Key words Bessel functions of matrix argument, matrix cones, Bessel functions associated with root systems, asymptotics, radial random walks, laws of large numbers, large deviations, Bessel hypergroups MSC (2010) 60B10, 33C67, 33C10, 43A05, 60F15, 60F10, 60F05, 44A10, 43A62For a fixed probability measure ν ∈ M 1 ([0, ∞[) and any dimension p ∈ N there is a unique radial probability measure νp ∈ M 1 (R p ) with ν as its radial part. In this paper we study the limit behavior of S p n 2 for the associated radial random walks (Sn ) n ≥0 on R p whenever n, p tend to ∞ in some coupled way. In particular, weak and strong laws of large numbers as well as a large deviation principle are presented.In fact, we shall derive these results in a higher rank setting, where R p is replaced by the space of p × q matrices and [0, ∞[ by the cone Πq of positive semidefinite matrices. All proofs are based on the fact that in this general setting the (S p k ) k ≥0 form Markov chains on Πq whose transition probabilities are given in terms Bessel functions Jμ of matrix argument with an index μ depending on p. The limit theorems then follow from new asymptotic results for the Jμ as μ → ∞.

A Limit Relation for Dunkl-Bessel Functions of Type A and B

“…More precisely, we shall prove that under suitable normalization of the arguments, the Bessel function J B (k 1 ,k 2 ) of type B q converges to the Bessel function J A k 2 of type A q−1 where the multiplicity parameter k 2 (on the roots ±e i ± e j ) is fixed and k 1 (on the roots ±e i ) tends to infinity. The obtained estimate is optimal for small arguments, but for large arguments it is weaker than (1) and the corresponding result in [14] for Bessel functions of matrix argument. This is due to the fact that the proofs of (1) and its matrix version in [14] rely on some explicit integral representation of the Bessel functions which is as far not available for Dunkl-type Bessel functions in general.…”

“…The obtained estimate is optimal for small arguments, but for large arguments it is weaker than (1) and the corresponding result in [14] for Bessel functions of matrix argument. This is due to the fact that the proofs of (1) and its matrix version in [14] rely on some explicit integral representation of the Bessel functions which is as far not available for Dunkl-type Bessel functions in general. Nevertheless, our limit result should be of some interest in its own.…”

“…We first recapitulate the necessary facts from Dunkl theory. We shall not go into details but refer the reader to [3,11] and [14] for more background. For multivariable hypergeometric functions, see e.g.…”

“…In these cases, the Bessel functions of type B are closely related with Bessel functions on the matrix cones Π N = Π N (F) of positive semidefinite N ×N matrices over F = R, C, H as explained in [12]. Using this connection, we shall derive Proposition 2 from a corresponding result for Bessel functions on matrix cones in [14]. We first recapitulate some facts about Bessel functions of matrix argument.…”

Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

“…We point out that our convergence results with error bounds may serve as a basis to derive central limit theorems for random walks on the Grassmannians G p,q (F) when for fixed rank q, the time parameter of the random walks as well as the dimension parameter p tend to infinity in a coupled way. For results in this direction we refer to [RV3], [V2].…”