I have i.i.d. random variables with following distribution:
$$ P(\xi_i =1) = p_1, \ P(\xi_i = 0) = p_0, \ P(\xi_i = -1) = p_{-1}; \quad S_n = \sum^n_{i=1}\xi_i.$$
I am interested in probability of $( S_n ) _{n=1}^{\infty}$ reaching some level $-L$ before reaching $L$.
I know the results for standard random walk with only two possible steps: -1 and 1. So my idea is to find the corresponding probability for this adjusted random walk:
$$ P(\xi'_i =1) = \frac{p_1}{p_1 + p_{-1}}, \ P(\xi'_i =-1) = \frac{p_{-1}}{p_1 + p_{-1}}; \quad S'_n = \sum^n_{i=1}\xi'_i.$$
And prove that it is the same as probability for my original walk.
I have found an answer for the "adjusted" walk but I am stuck with this proof. Any suggestions would be much appreciated.