ON RANDOM WEIGHTED SUM OF POSITIVE SEMI-DEFINITE MATRICES

Abstract

Let $A_1, \dots, A_n$ be fixed positive semi-definite matrices, i.e. $A_i \in \mathbb{S}_p^{+}(\mathbf{R})$ $\forall i \in \{1, \dots, n\}$ and $u_1, \dots, u_n$ are i.i.d. with $u_i \sim \mathcal{N}(1, 1)$. Then, the object of our interest is the following probability

$$\mathbb{P}\bigg(\sum_{i=1}^n u_i A_i \in \mathbb{S}_p^{+}(\mathbf{R})\bigg).$$

In this paper we examine this quantity for pairwise commutative matrices. Under some generic assumption about the matrices we prove that the weighted sum is also positive semi-definite with an overwhelming probability. This probability tends to $1$ exponentially fast by the growth of number of matrices $n$ and is a linear function with respect to the matrix dimension $p.$