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Proceedings of the Yerevan State University, series Physical and Mathematical Sciences, 2020, Volume 54, Issue 2, Pages 96–100
DOI: https://doi.org/10.46991/PYSU:A/2020.54.2.096
(Mi uzeru710)
 

Mathematics

On random weighted sum of positive semi-definite matrices

T. V. Galstyan, А. G. Minasyan

Yerevan State University
References:
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 $p$.
Received: 27.02.2020
Revised: 20.05.2020
Accepted: 17.08.2020
Document Type: Article
MSC: 60A05; 65C50
Language: English
Citation: T. V. Galstyan, А. G. Minasyan, “On random weighted sum of positive semi-definite matrices”, Proceedings of the YSU, Physical and Mathematical Sciences, 54:2 (2020), 96–100
Citation in format AMSBIB
\Bibitem{GalMin20}
\by T.~V.~Galstyan, А.~G.~Minasyan
\paper On random weighted sum of positive semi-definite matrices
\jour Proceedings of the YSU, Physical and Mathematical Sciences
\yr 2020
\vol 54
\issue 2
\pages 96--100
\mathnet{http://mi.mathnet.ru/uzeru710}
\crossref{https://doi.org/10.46991/PYSU:A/2020.54.2.096}
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