stability of stochastic sequence

Hi,

I would like to have a criterion for the stability (my definition of stability is given below) of the recurrence relation

Y(n+1)=(lambda+mu*N)*(1.95*Y(n)-0.95*Y(n-1))

with arbitrary starting values Y0 and Y1. In the equation above, lambda and mu are real parameters, and N is a random number sampled from a normal distribution with mean 0 and variance 1.

If mu=0 (the deterministic case), the stability (the fact that Y(n) goes to zero as n tends to infinity) is determined by the characteristic polynomial

xi^2=lambda*(1.95*xi-0.95),

of which both zeros should lie within the unit circle. This immediately leads to the stability condition lambda \in (-0.345...,1).

My question is now what happens if mu is not equal to zero. I then want to consider mean square stability, which means that the sequence is stable if

lim_{n \rightarrow \infty} E[Y(n)^2] = 0

(with E the mean of the distribution Y). So I am interested in a condition in terms of mu and lambda for which the sequence is mean square stable. Any of my attempts before failed because I get stuck with terms containing Y(n)*Y(n-1), of which I fail to determine the mean (as these numbers are correlated).

Any ideas how to solve my problem?

Thanks!
Christophe
.

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