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Principle Seminar of the Department of Probability Theory, Moscow State University
December 14, 2016 16:45–17:45, 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

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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 $n\to \infty$.
The sufficient conditions for the law of large numbers for the sequence of compositions of independent random semigroups are obtained. The examples of violation of the law of large numbers for such semigroups are given.

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