I have very little "hands-on" experience with probability, but here is my context:
I was looking at the random Fibonacci sequence: $$f_0=f_1=1, f_n=f_{n-1}+Xf_{n-2}$$
where $X$ is chosen randomly and uniformly as either $-1$ or $1$. It has been shown that this sequence always diverges exponentially.
Similarly, we can construct a trivial random sequence which always converges. $f_0=1, f_n=Xf_{n-1}$, where $X$ is chosen randomly and uniformly as either $.8$ or $1.2$.
So here is my not-well-defined question: could we construct a random sequence that sometimes converges and sometimes diverges, each with non-zero probability?
I don't have a good way of defining a random sequence. Naively, we can say that $f_n$ must be equal to a function of some of the previous terms as well as some number of random variables.
I realize that: 1. most examples will probably be degenerate/"edge" cases, and 2. an answer of "no, such sequences can't exist" is hard to prove given the vagueness of the problem.
Yes we could. For an example, consider a random walk $(f_n)$ to the nearest neighbor on the integer line with a drift to $+\infty$, stopped at $0$. That is, $f_0\geqslant1$, if $f_n\geqslant1$ then $f_{n+1}=f_n+g_n$ where $(g_n)$ is i.i.d. such that $P(g_1=+1)=p$, $P(g_1=-1)=1-p$ for some $p$ in $(\frac12,1)$, and if $f_n=0$ then $f_{n+1}=0$. Then, for every positive $f_0$, either $f_n=0$ for every $n$ large enough or $f_n\to+\infty$, and each event happens with nonzero probability.