$X_1,X_2,X_3,\ldots$ 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 money being partially invested someplace again and again with transaction fee each time (dependent on the current money amount).
Can I prove this or something similar to
$$Y_n\to\infty\,\, \textrm{a.s.}\,\,\,\,\textrm{if}\,\,\,\, E[\log(1+tX_1-f_i)] > 0\textrm{ for all $i$} \,\,\,\,\,\textrm{}$$
and
$$Y_n\to 0\,\, \textrm{a.s.}\,\,\,\,\textrm{if}\,\,\,\, E[\log(1+tX_1-f_i)] < 0\textrm{ for all $i$} \,\,\,\,\,\textrm{}$$
given some reasonable conditions on the $f$'s? If so, how to prove it and what are those conditions?
If this is possible I imagine those expectations would have to be conditional expectations since the $f_n$s depend on $Y_{n-1}$s, but I don't know how to start.
You're going to need another condition on your functions $f_i$, because for example they may map very small values to $10$ and still result in negative expectation in your proposed condition. Perhaps your functions need to be increasing.