I'm trying to solve problem 27 from Chapter XIV An Introduction to Probability Theory Volume I by William Feller, http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/An-Introduction-to-probability-Theory-by-William-Feller.pdf:
- In a three-dimensional symmetric random walk the particle has probability one to pass infinitely often through any particular line $x=m, y=n.$ (Hint: See Problem 5)
Here is problem 5
- Consider a random walk in which the particle starts at the origin and moves a unit step to the right or left, or stays at its present position with probabilities $\alpha, \beta, \gamma$, respectively ($\alpha+\beta+\gamma=1)$. Let $A,B>0$. Find $P(\text{the random walk reaches } A \text{ before } -B)$ and $E(\text{number of steps until the particle leaves } (-B,A)).$
I am really stuck here. I don't really know how problem 5 factors into this problem at all.
I am pretty sure that $$P(\text{the random walk reaches } A \text{ before } -B)=\frac{r^B-1}{r^{A+B}-1},$$ where $r=\frac{\beta}{\alpha}$, and $$ E(\text{number of steps until the particle leaves } (-B,A))=\frac{B}{1-2p}-\frac{A+B}{1-2p}\frac{r^B-1}{r^{A+B}-1}, $$ again where $r=\frac{\beta}{\alpha}$.
I appreciate any help with solving 27.
Thanks!