I'm interested in dealing with limit values for a linear recurrence relation, which is complicated slightly by stochastic transformations.
Here is the specific question. For $n<0$, let $a_n=a_n^{\ast}=0$. Let $a_0=a_0^{\ast}=a_1=a_1^{\ast}=1$. Then for $n\geq 1$ define:
$$a_{n+1}= a^{\ast}_{n}+\sum_{j\geq 1}4ja^{\ast}_{n-j} = a^{\ast}_n +4a^{\ast}_{n-1}+8a^{\ast}_{n-2} + 12 a^{\ast}_{n-3} \dots $$
Then $a_{n+1}^{\ast}$ is defined to be the value of a Poisson random variable with mean $a_{n+1}$. We are interested in the values $a_{n}^{\ast}$ for large $n$.
Okay, so in the absence of the stochastic element, if it were simply the case that $a_{n}^{\ast}=a_n$ and this was a standard recurrence relation, since we would then have $a_n=a_{n-1}+\sum_{j\geq 1}4ja_{n-j-1}$ we could rewrite, for $n>0$:
$$a_{n+1}= 2a_n + 3n_{n-1} + 4\sum_{j\geq 2}a_{n-j}.$$
Given that $a_{n-2}=2a_{n-3}+3a_{n-4}+4\sum_{j\geq 2} a_{n-j-3}$, this in turn then gives, for $n>0$:
$$a_{n+1}= 2a_n + 3a_{n-1} + 5a_{n-2}+2a_{n-3}+a_{n-4}.$$
Since the characteristic polynomial $x^5-2x^4-3x^3-5x^2-2x-1=0$ has a largest root $\alpha \approx 3.38298$, it follows that for some constant $\rho>0$: $$ \lim_{n\to \infty} \frac{a_n}{\rho\cdot \alpha^n} = 1. $$
OKAY.. so.. for the stochastic version, what I suppose happens is that with probability 1, we have $a^{\ast}_n/\alpha^n$ tending to a limit. One could presumably go in and prove it from first principles, but I expect this follows fairly directly from known theorems? Many thanks!