Are there such things as recurrence equations with random variable coefficients. For example, $$W\_n=W\_{n-1}+F\cdot W\_{n-1}$$ where $F$ is a random variable. I tried to see if I could make sense of it using the simplest possible case of $F$ being a uniform discrete random variable on 2 points but I didn't get far because even if the initial data is not random the succeeding terms in the sequence are and each term seems to live on a different space. I couldn't figure out what space $W\_n$ and $W\_{n-1}$ should live on. A google search turned up nothing for the obvious keywords "random recurrence equation".


Edit in response to Alekk's answer: More specifically suppose I wanted to find the probability $P(W\_{200}>3000)$. Is there a way to compute the distribution of $W\_{200}$ explicitly given the distribution of $F$ and some non-random initial data $W\_0$?


Edit: $F$ does not depend on $n$ and to make things even more explicit lets say $F$ has the distribution $P(F=2f)=\dfrac{1}{2}, P(F=-f)=\dfrac{1}{2}$.



linear recurrence relations with random coefficients
So this is the product of IID random variables $1+F\_n$, so you could take logarithms and do the more conventional sums of IID random variables $\log(1+F\_n)$. Perhaps the logarithms are complex numbers.


