Distributions of local times of a single excursion of 1D random walk

Question

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq \frac{1}{2}$. For every integer $y \in \mathbb{Z}$, call $Y_y$ the random variable returning the number of visits at $y$ before the random walker hits the origin.

How can I characterize the distribution of $Y_y$ for general $y$?

Answer 1

If $y=0$, then $Y_y=0$ or $Y_y=1$, depending on the convention. If $y\gt0$, then:

• $P(Y_y\gt0)=pP_1(T_y\lt T_0)$
• $P(Y_y\gt n+1\mid Y_y\gt n)=P_y(T_y\lt T_0)=pP_{y+1}(T_y\lt\infty)+(1-p)P_{y-1}(T_y\lt T_0)$

Similar recursions hold when $y\lt0$. If one is able to compute the various probabilities $P_a(T_b\lt T_c)$ involved (and one should be), these considerations solve the problem.