Non-causal strictly stationary solutions of random recurrence equations

Let (Mn,Qn)n∈N(Mn,Qn)n∈N be an i.i.d. sequence in R2R2. Much attention has been paid to causal strictly stationary solutions of the random recurrence equation Xn=MnXn−1+QnXn=MnXn−1+Qn, n∈Nn∈N, i.e. to strictly stationary solutions of this equation when X0X0 is assumed to be independent of (Mn,Qn)n∈N(Mn,Qn)n∈N. Goldie and Maller (2000) gave a complete characterisation when such causal solutions exist. In the present paper we shall dispose of the independence assumption of X0X0 and (Mn,Qn)n∈N(Mn,Qn)n∈N and derive necessary and sufficient conditions for a strictly stationary, not necessarily causal solution of this equation to exist.