Variance inequalities for quadratic forms with applications

Abstract

We obtain variance inequalities for quadratic forms of weakly dependent random variables with bounded fourth moments. We also discuss two application. Namely, we use these inequalities for deriving the limiting spectral distribution of a random matrix and estimating the long-run variance of a stationary time series.

Limiting spectral distribution for large sample covariance matrices with graph-dependent elements

“…The most general conditions imposed on x p ensure that the quadratic forms x ⊤ p A p x p weakly concentrate around their expectations up to an error term o(p) with probability 1 − o (1), where A p ∈ C p×p is an arbitrary matrix with the spectral norm A p 1. These conditions were studied in [2], [8], [16], [22], [28], [29], and [30]. As shown in [29], the weak concentration property for specific quadratic forms of x p gives necessary and sufficient conditions for the Marchenko-Pastur theorem [18].…”

Marchenko-Pastur law for a random tensor model

“…In particular, [30] gives necessary and sufficient conditions for the MP theorem in terms of concentration of certain quadratic forms. For various models of data, the concentration properties are established in [4], [5], [20], [23], [29], [31], [32], among others.…”