Operator Random Walks and Quantum Oscillator

We investigate asymptotic properties of a sequence whose elements are probability distributions which correspond to compositions of independent random operators. We prove the following statements: 1) a sequence whose elements are the expected values which corresponds to compositions of independent identically distributed random semigroups converges to a semigroup which is generated by the expected value of the random generator; 2) an analog of the law of large numbers for a sequence whose elements are compositions of independent identically distributed random semigroups; 3) a sequence whose elements are compositions of random shift operators on position space converges to a semigroup generated by the Laplace operator; 4) a sequence whose elements are compositions of random shift operators on position space and random shift operators in momentum space converges to a semigroup generated by the Hamiltonian of the quantum oscillator; 5) a sequence whose elements are compositions of random shift operators on phase space converges to a semigroup generated by the Hamiltonian of the quantum oscillator.