Solving this random variable problem

Question

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first?

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$.

Also $t\in(0,1)$.

$f_1>0$ is a positive constant and $f_2,f_3,f_4\ldots$ are positive functions of one variable.

Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like

$$Y_1 = (1+tX_1) - f_1$$ $$Y_n = Y_{n-1}(1+tX_n) - f_n(Y_{n-1})$$

Imagine this is money being partially invested someplace again and again with transaction fee each time (dependent on the current money amount).

Can I prove that

$$Y_n\to\infty\,\, \textrm{a.s.}\,\,\,\,\textrm{if and only if}\,\,\,\, E[\log(1+tX_1)] > 0 \,\,\,\,\,\textrm{}$$

given some reasonable conditions on the $f$'s? If so, what are those conditions?

Answer 1

Here is one set of conditions.

Assume that $f_1\leqslant1-t$ and that, for every $n\geqslant2$, there exists some $a_n\leqslant1-t$ such that $f_n(x)\leqslant a_nx$ for every $x\geqslant0$. Consider $\liminf\limits_{n\to\infty}a_n=a$ and assume that $E(\log(1+tX_1-a))\gt0$ (note that $1+tX_1\gt a$ almost surely).

Then $Y_n\to+\infty$ almost surely.