Let $W_0^+,W_1^+,...$ i.i.d random variables on $[0,1]$ and $W_0^-:=1-W_0^+,...$.
Suppose that X, given W, is a random walk on $\mathcal{N}_0$ with transition probabilities $p_i(x,x+1)=W_i^+$, $p_i(x,x-1)=W_i^-$ and, if $x=0$, the process cannot go to to the negative integers but it stays in $0$.
How I can show that X is null recurrent if $E[\log{\frac{W_0^-}{W_0^+}}]=0$?
I know that X is recurrent from Solomon (1975). To prove the null recurrent property I tried to obtain some recurrent equations with no success because, at the end, I get an expected value that involves both the "$W$"s and some other expectation conditioned on those "$W$"s.
Please, let me know if more context is needed. Thanks for the help.