Geodesic
On random weighted sum of positive semi-definite matrices
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 54 (2020) no. 2, pp. 96-100 
T. V. Galstyan; А. G. Minasyan. On random weighted sum of positive semi-definite matrices. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 54 (2020) no. 2, pp. 96-100. http://geodesic.mathdoc.fr/item/UZERU_2020_54_2_a2/
@article{UZERU_2020_54_2_a2,
     author = {T. V. Galstyan and {\CYRA}. G. Minasyan},
     title = {On random weighted sum of positive semi-definite matrices},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
     pages = {96--100},
     year = {2020},
     volume = {54},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZERU_2020_54_2_a2/}
}
TY  - JOUR
AU  - T. V. Galstyan
AU  - А. G. Minasyan
TI  - On random weighted sum of positive semi-definite matrices
JO  - Proceedings of the Yerevan State University. Physical and mathematical sciences
PY  - 2020
SP  - 96
EP  - 100
VL  - 54
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UZERU_2020_54_2_a2/
LA  - en
ID  - UZERU_2020_54_2_a2
ER  - 
%0 Journal Article
%A T. V. Galstyan
%A А. G. Minasyan
%T On random weighted sum of positive semi-definite matrices
%J Proceedings of the Yerevan State University. Physical and mathematical sciences
%D 2020
%P 96-100
%V 54
%N 2
%U http://geodesic.mathdoc.fr/item/UZERU_2020_54_2_a2/
%G en
%F UZERU_2020_54_2_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $p$.

[1] B. Efron, “Bootstrap Methods: Another Look at the Jackknife”, Ann. Statist., 1979, 1–26 | DOI | MR | Zbl

[2] M. R. Chernick, W. González-Manteiga, R. M. Crujeiras, E. B. Barrios, “Bootstrap Methods”, International Encyclopedia of Statistical Science, ed. Lovric M., Springer, Berlin–Heidelberg, 2011, 169–174 | DOI

[3] W. González-Manteiga, J. M. Prada Sánchez, J. Romo, The Bootstrap. A Review. Working Paper, Universidad Carlos III de Madrid. Departamento de Economia, 1992 | DOI | MR

[4] I. A. Ibragimov, R. Z. Khasminskii, Statistical Estimation – Asymptotic Theory, Springer, N. Y., 1981 | DOI | MR | Zbl

[5] A. Minasyan, “Alternating Least Squares in Generalized Linear Models”, Contemp. Mathemat. Anal., 2019, 302–312 | DOI | MR | Zbl

[6] A. V. Minasyan, Bootstrap Confidence Sets for Generalized Linear Models, Master's Thesis, Skolkovo Institute Science and Technology (Skoltech), Moscow, 2018

[7] K. T. Conrad, The Minimal Polynomial and Some Applications, Tech. report. Dept. of Mathematics, Univ. of Connecticut, 2008