A note on the comparison of stationary laws of Markov processes

Let {Xn}, {Yn} be Markov process on k, satisfying Xn+1 = T1(Xn)+Zn, Yn+1 = T2(Yn)+Zn, where {Zn} are i.i.d random variables. Let [mu]X resp. [mu]Y be the stationary distributions of {Xn}resp. {Yn}. We introduce an order relation for probabilities measuring the degree of concentration around zero and derive a result connecting this degree of concentration with properties of the functions Ti and the distribution of {Zn}. Our theorem generalizes a known result for the univariate case which was given by Högnäs (1986).