  We present a procedure for averaging one-parameter random unitary groups and
random self-adjoint groups. Central to this is a generalization of the notion
of weak convergence of a sequence of measures and the corresponding
generalization of the concept of convergence in distribution. The convergence
is established in determination of the sequence of compositions of independent
random transformations. When sequences of compositions of independent random
transformations of the shift by the Euclidean vector in space, the results
obtained coincide with the central limit theorem for the sums independent
random vectors. The results are applied to the dynamics of quantum systems
arising random quantization of the classical Hamiltonian system.
