I am studying a simple random growth process: $$ X_{n+1} = A_{n}\, X_{n} + Y_{n} \\ Y_{n+1} = B_{n}\, Y_{n} $$ with both $A_{n}$ and $B_{n}$ independent lognormal random variables.
In particular, I am interested in the choice of parameters $\mathbb{E}[\log(A_{n})] > \mathbb{E}[\log(B_{n})]$, for which the process has infinite memory.
I would like to prove that $\mathbb{E}[Y_{n-1}|X_{n}]$ is a non-decreasing function of $X_{n}$ (for some generic initial condition, for example for $X_0$ and $Y_0$ independently distributed).
It seems an almost trivial statement, but the proof appears much less trivial.