A law of large numbers and central limit theorem for the logarithm of an autoregressive process with a stationary driving sequence

Let (ξ,η,ν)={(ξi,ηi,νi)}i∈Z(ξ,η,ν)={(ξi,ηi,νi)}i∈Z be a stationary and ergodic sequence in R3R3. Consider the autoregressive process defined by, R0(ξ,η,ν)=η0R0(ξ,η,ν)=η0 and Rn(ξ,η,ν)=ξnRn−1(ξ,η,ν)+ηnνn⋅…⋅ν1,n≥1Rn(ξ,η,ν)=ξnRn−1(ξ,η,ν)+ηnνn⋅…⋅ν1,n≥1. We give conditions under which limn→∞1nln|Rn(ξ,η,ν)| exists a.s. and at the same time, Eln|ξ0|