Convolutions of random measures on compact groups

Let G be a compact group and M₁(G) be the convolution semigroup of all Borel probability measures on G with the weak topology. We consider a stationary sequence {μ_n}_n=-∞^+∞ of random measures μ_n=μ_n(ω) in M₁(G) and the convolutions {Mathematical expression} and {Mathematical expression} We describe the sets A_m⁺(ω) and A_n⁺(ω) of all limit points of v_m,n(ω) as m→-∞ or n→+∞ and the set A_∞(ω) of its two-sided limit points for typical realizations of {μ_n(ω)}_n=-∞^+∞. Using an appropriate random ergodic theorem we study the limit random measures ρ_n^(±)(ω)=lim_k→∞ α_n^(±k)(ω).