c Allerton Press, Inc., 2016.
ISSN 1066-369X, Russian Mathematics , 2016, Vol. 60, No. 10, pp. 72–76. c V.Zh. Sakbaev, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 10, pp. 86–91.
Original Russian Text On the Law of Large Numbers for Compositions
of Independent Random Semigroups
V. Zh. Sakbaev1*
(Submitted by A.M. Bikchentaev)
1
Steklov Mathematical Institute of Russian Academy of Sciences
ul. Gubkina 8, Moscow, 119991 Russia
Received March 24, 2016
Abstract—We study random linear operators in Banach spaces and random one-parameter semigroups of such operators. For compositions of independent random semigroups of linear operators
in the Hilbert space we obtain sufficient conditions for fulfilment of the law of large numbers and give
examples of its violation.
DOI: 10.3103/S1066369X16100121
Keywords: law of large numbers, random map, random semigroup, Chernoff theorem.
1. Introduction. In this work we investigate random operators, random one-parameter semigroups
and their iterations. For a sequence of compositions of n independent equidistributed random semigroups of operators we study asymptotic behavior of difference between a composition and its mean
value as n → ∞.
n
ηk , n ∈ N with independent usual random variables ηn , n ∈ N the law
For the sequence Sn = n1
k=1
of large number asserts that P ({|Sn − M η| > ε}) → 0 for n → ∞ and for all ε > 0, where M η is mean
value of random variable ηk . In this paper we put the question about asymptotic behavior of a sequence
1/n
1/n
compositions Un = Un ◦ · · · ◦ U1 , n ∈ N, where {Un } are independent random variables with
values in a set of one-parameter semigroups of linear operators in the Hilbert space H.
We will speak that for sequence of compositions {U(n)} of random semigroups with values in Banach
space of operator-valued functions X the law of large numbers is fulfilled, if the probability that the
difference between the composition U(n) and its mean value exceed (by norm of space X) some positive
number, tends to zero as n → ∞. We will speak that for sequence of compositions {U(n)} of random
semigroups with values in topological linear space of operator-valued functions Y the law of large
numbers is fulfilled, if for any semi-norm p in the set S, that defines a topology of Y , the probability
that the difference between the composition U(n) and its mean value exceed (by semi-norm p) some
positive number, tents to zero at n → ∞.
The study of compositions of random linear transformation in Banach space is interesting for
noncommutative theory of probability, for dynamic systems theory, theory of stochastic differential
equations.
In papers [1–4] they study the random dynamical systems with discrete time as the sequences of
compositions n of independent random transformations of phase space. In these papers they also study
invariant measures, decomposition of dynamics on determinate and stochastic components, ergodic
properties, attractors. In papers [3, 5, 6] they obtain sufficient conditions such that distributions of
logarithms of Lyapunov’s indices of compositions of n random linear operators tend to normal as
n → ∞. In [7] the limit distributions of spectral radius for compositions of random linear operators
are investigated.
In the present work we investigate random semigroups of linear transformations of Banach space
introduced in [8] and studied in papers [9, 10] and obtain conditions on random semigroups of operators
*
E-mail: [email protected].
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