Open AccessStochastic models in artificial intelligence development
Author(s) -
Oksana Kyrychenko,
AUTHOR_ID,
Igor V. Malyk,
С. Е. Остапов,
AUTHOR_ID,
AUTHOR_ID
Publication year - 2021
Publication title -
vìsnik. serìâ fìziko-matematičnì nauki/vìsnik kiì̈vsʹkogo nacìonalʹnogo unìversitetu ìmenì tarasa ševčenka. serìâ fìziko-matematičnì nauki
Language(s) - English
Resource type - Journals
eISSN - 2218-2055
pISSN - 1812-5409
DOI - 10.17721/1812-5409.2021/2.7
Subject(s) - eigenvalues and eigenvectors , mathematics , independence (probability theory) , random matrix , convergence (economics) , matrix (chemical analysis) , stochastic matrix , cluster analysis , statistics , markov chain , economics , composite material , economic growth , physics , materials science , quantum mechanics
In this paper, we consider some properties of stochastic random matrices of large dimensions under conditions of independence of matrix elements or under conditions of independence of rows (columns). The main properties of stochastic random matrices spectrum are analyzed and the result of convergence to 0 is proved of almost all eigenvalues. Also, the application of these results to clustering problems and selection of the optimal number of clusters is considered. Note that the results obtained in this work are consistent with the Marchenko - Pastur theorem on the asymptotic distribution of eigenvalues of random matrices with independent elements. The results proved in this paper can be interpreted as a law of large numbers and will be used in the study of the asymptotic behavior of the maximum.