Limit theorems for radial random walks on p × q-matrices as p tends to infinity
✍ Scribed by Margit Rösler; Michael Voit
- Publisher
- John Wiley and Sons
- Year
- 2010
- Tongue
- English
- Weight
- 213 KB
- Volume
- 284
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
For 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 → ∞.