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Principle Seminar of the Department of Probability Theory, Moscow State University
December 14, 2016 16:45, Moscow, MSU, auditorium 12-24

On the law of the large numbers for the composition of random operators and semigroups

V. Zh. Sakbaevab

a Moscow Institute of Physics and Technology
b Peoples Friendship University of Russia

Abstract: The random linear operators in Banach spaces, one-parameter semigroups of such operators and its iterations are studied. The asymptotic of deviation of compositions of $n$ independent identically distributed random operators from its mean value for $n\to \infty$ is studied.
The law of the large numbers for the sequence $S_n={1\over n}\sum\limits_{k=1}^n\eta _k, n\in N$, of the sums of independent real valued random variables $\eta _n, n\in N$, states that $P(\{ |S_n-MS_n |>\epsilon \})\to 0$ for $n\to \infty$ for any $\epsilon >0$ where $MS_n$ is the mean value of random variable $\eta$ and $P(\{ |S_n-MS_n |>\epsilon \})$ is the probability of the event that the deviation of random variable $S_n$ from its mean value is more than $\epsilon$. For the sequence $\{ {U}_n\}$ of independent random variables with values in the set of one-parametric semigroups of linear operators in some Hilbert space $H$ the asymptotic behavior of the sequence of averaged compositions $U(n)=U_n^{1\over n}\circ ...\circ {U}_1^{1\over n}, n\in N$ is investigated.
The sequence of averaged compositions $\{ U(n)\}$ of the independent random semigroups with values in some Banach (locally convex) space of operator valued functions $X$ is said to satisfy the large numbers law if the probability of the event that the deviation of composition $U(n)$ from its mean value in the norm of the space $X$ (in any seminorm from the family of seminorms generating the topology of the space $X$) is more than $\epsilon >0$ tends to zero for $n\to \infty$.