Limit distribution for compositions of random operators

Limit theorems for compositions of independent linear operators acting in a finite dimensional Euclidean space $E$ are studied. An example of application of the limit theorems to construction of equations corresponding to random independent affine transformations of a Euclidean space is explored.
It is known (see [1]) that the limit properties of distribution of the sum of random variables with values in the topological vector spaces can be described by limit theorems. In particular, the law of large numbers describes the convergence in probability of the sequence of averaged sum of independent identically distributed (iid) random vector valued variables to the limit of the mean value of the sum. The central limit theorem gives the conditions of the convergence in distribution for the sequence of averaged sum of iid random vector valued variables to the Gaussian random vector.
We study the sequence of compositions of iid random variables with values in the Banach algebra of bounded linear operators $B(H)$ acting in the separable Hilbert space $H$. In the commutative case of operators of an argument shift on a random vector the limit distribution of averaged composition can be described by the limit theorems for the sum of vector valued variables. Some results on the LLN and CLT for the averaged composition of independent random matrices or linear operators was obtained in [2, 3, 4]. We obtain the analogs of LLN and CLT for the sequence of compositions of iid random semigroups or $B(H)$-valued random processes with non-commutative values.