XXT matrices with independent entries

Article Type

Research Article

Publication Title

Alea (Rio de Janeiro)

Abstract

Let S = XXT be the (unscaled) sample covariance matrix where X is a real p × n matrix with independent entries. It is well known that if the entries of X are independent and identically distributed (i.i.d.) with enough moments and p/n → y ≠ 0, then the limiting spectral distribution (LSD) of (Formula Presented) converges to a Marčenko-Pastur law. Several extensions of this result are also known. We prove a general result on the existence of the LSD of S in probability or almost surely, and in particular, many of the above results follow as special cases. At the same time several new LSD results also follow from our general result. The moments of the LSD are quite involved but can be described via a set of partitions. Unlike in the i.i.d. entries case, these partitions are not necessarily non-crossing, but are related to the special symmetric partitions which are known to appear in the LSD of (generalised) Wigner matrices with independent entries. We also investigate the existence of the LSD of SA = AAT when A is the p × n symmetric or the asymmetric version of any of the following four random matrices: reverse circulant, circulant, Toeplitz and Hankel. The LSD of (Formula Presented) for the above four cases have been studied in (Bose et al., 2010) when the entries are i.i.d. We show that under some general assumptions on the entries of A, the LSD of SA exists and this result generalises the existing results of (Bose et al., 2010) significantly

First Page

75

Last Page

125

DOI

https://10.30757/ALEA.v20-05

Publication Date

1-1-2023

Comments

Open Access, Green

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