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Mathematics
On random weighted sum of positive semi-definite matrices
T. V. Galstyan, А. G. Minasyan Yerevan State University
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
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
Linking options:
https://www.mathnet.ru/eng/uzeru710 https://www.mathnet.ru/eng/uzeru/v54/i2/p96
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Abstract page: | 96 | Full-text PDF : | 43 | References: | 19 |
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