One of the more interesting problems that I've come across recently is to calculate the distribution of the last time a simple random walk is at $0.$

Let $X_1,X_2,\ldots,X_{2n}$ be independent, indentically distributed random variables such that $P(X_i = 1) = P(X_i = -1) = 1/2.$ Define $S_k = \sum_{i=1}^k X_i.$ Then, we have a path $$(0,0) \rightarrow (1,S_1) \rightarrow (2,S_2) \rightarrow \cdots \rightarrow (2n,S_{2n}).$$ Define the random variable $L_{2n} = \sup\{ k \leq 2n : S_k = 0\}.$ We want the distribution of $L_{2n}.$

Note that we have that \begin{equation} \mathbb{P}(S_{2n} = 0) = 2^{-2n}{2n \choose n} \end{equation} since we have $n$ positive steps and $n$ negative steps.

Let $\displaystyle N_{n,x} ={n \choose (n+x)/2}$ denote the number of paths from $(0,0)$ to $(n,x)$ since $(n+x)/2$ positive steps implies there are $(n-x)/2$ negative steps, and $(n+x)/2 - (n-x)/2 = x$. Note that $n + x$ must be even for this to be well defined. If $n + x$ is not even, then $N_{n,x} = 0$ since $x$ must have the same parity as $n.$ First, we prove the reflection principle.

Reflection Principle

If $x,y > 0,$ the number of paths that from $(0,x)$ to $(n,y)$ that are $0$ at some time, that is, they touch the $x$-axis is equal to the total number of paths from $(0,-x)$ to $(n,y),$ which is $N_{n,y+x}.$ Therefore, the number of paths from $(0,x)$ to $(n,y)$ that do not touch $0$ is $N_{n,|y-x|} - N_{n,y+x}.$

We can establish a one-to-one correspondence between the set $A$, the paths from $(0,x)$ to $(n,y)$ that are $0$ at some time and the set $B$ the paths from $(0,-x)$ to $(n,y)$.

Consider any path $P$ in $A$. $P$ must include the point $(m,0),$ where $0 < m < n$. Fix $m$ to be the greatest such integer. We construct a path $Q_1$ from $(0,-x)$ to $(m,0)$ by going in the opposite direction as $P.$ We construct a path $Q_2$ from $(m,0)$ to $(n,y)$ by mirroring $P$. Thus, we have that $Q = Q_1 \cup Q_2 \in B.$

Now consider any path $Q$ in $B$. Since paths are continuous, $Q$ must cross the $x$-axis, so $Q$ includes a point $(m,0)$, where $0 < m < n.$ Fix $m$ to be the greatest such integer. We construct $P_1,$ a path from $(0,x)$ to $(m,0)$ by going in the opposite direction as $Q$. We construct $P_2$ by mirroring $Q$. Thus, we have that $P = P_1 \cup P_2 \in A.$

So, we have established a one-to-one correspondence, and therefore, we have proven $|A| = |B|.$

Symmetry of Zeroes

$\mathbb{P}(S_1 \neq 0, S_2 \neq 0,\ldots,S_{2n} \neq 0) = \mathbb{P}(S_{2n} = 0).$

First note that $$ \mathbb{P}(S_1 \neq 0,\ldots,S_{2n} \neq 0) = \mathbb{P}(S_1 > 0,\ldots,S_{2n} > 0) + \mathbb{P}(S_1 < 0,\ldots,S_{2n} < 0) $$ since we can never have the path touch $0.$ Also note that the two terms are equal, so \begin{equation} \mathbb{P}(S_1 \neq 0, S_2 \neq 0,\ldots,S_{2n} \neq 0) = 2\mathbb{P}(S_1 > 0, S_2 > 0,\ldots,S_{2n} > 0). \end{equation} Now, note that \begin{equation} \mathbb{P}(S_1 > 0, S_2 > 0,\ldots,S_{2n} > 0) = \sum_{r=1}^{n}\mathbb{P}(S_1 > 0, S_2 > 0,\ldots,S_{2n} = 2r) \end{equation} since we have taken an even number of steps.

To calculate $\mathbb{P}(S_1 > 0, S_2 > 0,\ldots,S_{2n} = 2r),$ we note that $X_1 = 1$ since $S_1 > 0$. Then, the number of paths from $(1,1)$ to $(2n,2r)$ that do not touch $0$ by the Reflection Principle is $N_{2n-1,2r-1} - N_{2n-1,2r+1}$. Thus, \begin{align*} \mathbb{P}(S_1 > 0, S_2 > 0,\ldots,S_{2n} = 2r) &= \left(\frac{1}{2}\right)\left(\frac{1}{2^{2n-1}}\right)\left(N_{2n-1,2r-1} - N_{2n-1,2r+1}\right) \\ &= \left(\frac{1}{2^{2n}}\right)\left(N_{2n-1,2r-1} - N_{2n-1,2r+1}\right). \end{align*}

So, we have that \begin{align*} \mathbb{P}(S_1 > 0, S_2 > 0,\ldots,S_{2n} > 0) &= \left(\frac{1}{2^{2n}}\right) \sum_{r=1}^n\left(N_{2n-1,2r-1} - N_{2n-1,2r+1}\right) \\ &= \frac{1}{2^{2n}}N_{2n-1,1} = \frac{1}{2^{2n}}{2n - 1 \choose n}\\ &= \frac{1}{2^{2n}}\frac{(2n-1)!}{n!(n-1)!} = \frac{1}{2}\frac{1}{2^{2n}}\frac{(2n)!}{n!n!} \\ &= \frac{1}{2}\mathbb{P}(S_{2n} = 0) \end{align*} by telescoping since $N_{2n-1,2n+1} = 0$ and substituting in the previous equations.

Recalling the earlier equation, we have the desired result, $$ \mathbb{P}(S_1 \neq 0, S_2 \neq 0,\ldots,S_{2n} \neq 0) = 2\mathbb{P}(S_1 > 0, S_2 > 0,\ldots,S_{2n} > 0) = \mathbb{P}(S_{2n} = 0). $$

Distribution of Last Zero Visit

Let $L_{2n}$ be the random variable whose value is the last time the simple random walk visited $0.$ Formally, $$L_{2n} = \sup\left\{2k : k \in \{0,\ldots,n\},~\sum_{i=1}^{2k} X_i = 0\right\}.$$ Then, we have that $$\mathbb{P}(L_{2n} = 2k) = 2^{-2n}{2k \choose k}{2n-2k \choose n-k}.$$

To see this, we have that \begin{align*} \mathbb{P}(L_{2n} = 2k) &= \mathbb{P}(S_{2k} = 0, S_{2k+1} \neq 0, \ldots,S_{2n} \neq 0) \\ &= \mathbb{P}(S_{2k} = 0, S_{2k+1} - S_{2k} \neq 0, \ldots,S_{2n} - S_{2k} \neq 0) \\ &= \mathbb{P}(S_{2k} = 0)\mathbb{P}(S_1 \neq 0, \ldots,S_{2n-2k} \neq 0) \\ &= \mathbb{P}(S_{2k} = 0)\mathbb{P}(S_{2n-2k} = 0), \end{align*} where the last equaility is by Symmetry of Zeroes. Thus, we have that $$\mathbb{P}(L_{2n} = 2k) = \mathbb{P}(S_{2k} = 0)\mathbb{P}(S_{2n-2k} = 0) =2^{-2n}{2k \choose k}{2n-2k \choose n-k}$$ as desired.

Conculusion

Let's look at what this distribution looks like when $n = 15.$ We have symmetric distribution about $15$, so the mean is $15.$ However, the distribution is U-shaped, and the most likely values are very far from the mean. Thus, to take an analogy from sports, if two evenly matched teams were to played against each other multiple times over the course of a season, the most likely scenario is for one team team to lead the other team the entire season. So, saying that team A led team B the entire season is an almost meaningless statistic.