Operator Approach to Weak Convergence of Measures and Limit Theorems for Random Operators

The generalized weak convergence of a sequence of measures is induced by the convergence of the linear operators generated by the measures. A corresponding generalization of the notion of convergence over a distribution is introduced. Generalized convergence over the distribution of a sequence of co...

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Bibliographic Details
Published inLobachevskii journal of mathematics Vol. 42; no. 10; pp. 2413 - 2426
Main Authors Orlov, Yu. N., Sakbaev, V. Zh, Shmidt, E. V.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.10.2021
Springer Nature B.V
Subjects
Algebra
Analysis
Boundary value problems
Central limit theorem
Composition
Convergence
Euclidean geometry
Euclidean space
Geometry
Linear operators
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Theorems
Transformations (mathematics)
random linear operator
random operator valued process
law of large numbers
central limit theorem
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Summary:The generalized weak convergence of a sequence of measures is induced by the convergence of the linear operators generated by the measures. A corresponding generalization of the notion of convergence over a distribution is introduced. Generalized convergence over the distribution of a sequence of compositions of independent random transformations is investigated. The connection between limit distributions and semigroups that solve initial-boundary value problems for evolution equations is established. In the case of a sequence of compositions of independent random transformations of the shift to a random vector of Euclidean space, the results obtained coincide with the central limit theorem for sums of independent random vectors.
ISSN:1995-0802
1818-9962
DOI:10.1134/S1995080221100188