Gambler’s ruin from harmonicity

For each $x\in E_N$, define a measure-valued row by $K_N(x,\cdot )=\{\begin{array}{ll}{\delta }_0,& x=0,\\ {\delta }_N,& x=N,\\ \frac{1}{2}{\delta }_{x-1}+\frac{1}{2}{\delta }_{x+1},& 1\le x<N\mathrm{.}\end{array}$ By [F2], each row is a probability measure; because $E_N$ is finite and discrete, the map $x\mapsto K_N(x,B)$ is measurable for each $B\subseteq E_N$. Thus $K_N$ is a probability kernel. By [F4], its transition matrix is exactly the one in the Statement, including the zero weights off the listed transitions.

Define boundary data $f_N(0)=0$, $f_N(N)=1$, and set $g_N(i)=i\mathrm{/}N$ on $E_N$. The function $g_N$ is bounded and agrees with $f_N$ on $A_N$. If $1\le i<N$, then [F4] and direct arithmetic give $P_N{g}_N(i)=\frac{1}{2}{g}_N(i-1)+\frac{1}{2}{g}_N(i+1)=\frac{(i-1)+(i+1)}{2N}=\frac{i}{N}=g_N(i)\mathrm{.}$ Thus $g_N$ satisfies the boundary and harmonic equations, with no interior equations required when $N=1$.

For each $i\in E_N$, take the canonical chain with kernel $K_N$ and initial law ${\delta }_i$, and write its law and expectation as ${\mathbb{P}}_i$ and ${\mathbb{E}}_i$. By [A1, F3], all these deterministic-start chains exist on the canonical path space and have the stated transition matrix.

Define the bounded path functional $H_N:E_N^{{\mathbb{N}}_0}\to \{0,1\}$ by $H_N(w)=1$ exactly when $1\le w_0<N$ and $w_s=\max \{w_0-s,0\}(1\le s\le N),$ and set $H_N(w)=0$ otherwise. It is measurable because it depends on finitely many coordinates in a finite discrete space. Under ${\mathbb{P}}_i$, for $1\le i<N$, the event $H_N=1$ specifies exactly $i$ left moves, each of probability $1\mathrm{/}2$, followed by the absorbing self-loop at $0$; hence [F5] gives ${\mathbb{E}}_i{H}_N(X_0,X_1,\dots )=2^{-i}\mathrm{.}$ For $i=0$ or $i=N$ the expectation is zero by the definition of $H_N$. Thus the canonical expectation function is $h_N(i)=\{\begin{array}{ll}{2}^{-i},& 1\le i<N,\\ 0,& i\in \{0,N\}\mathrm{.}\end{array}$

Put $A_N=\{0,N\}$ and $T=T_{A_N}$. By [F6], for every $m\ge 0$, ${\mathbb{E}}_i[H_N(X_m,X_{m+1},\dots )\mid {\mathcal{F}}_m]=h_N(X_m)\text{almost surely}\mathrm{.}$ On $\{T>mN\}$ the state $X_{mN}$ lies in $\{1,\dots ,N-1\}$. The event $H_N(X_{mN},X_{mN+1},\dots )=1$ then forces a visit to $0$ within the next $N$ steps, so $T\le (m+1)N$. Therefore [F6, F7] imply $\begin{array}{rl}{\displaystyle {\mathbb{P}}_i(T>(m+1)N)}& {\displaystyle ={\mathbb{E}}_i\text{\hspace{0.17em}⁣}[{\mathbf{1}}_{\{T>mN\}}{\mathbb{E}}_i[{\mathbf{1}}_{\{T>(m+1)N\}}\mid {\mathcal{F}}_{mN}]]}\\ {\displaystyle }& {\displaystyle \le (1-2^{-N}){\mathbb{P}}_i(T>mN)}\end{array}$ for every interior start; here $2^{-X_{mN}}\ge 2^{-N}$ on the survival event. Induction gives ${\mathbb{P}}_i(T>mN)\le (1-2^{-N}{)}^m$. Since $\{T=\mathrm{\infty }\}\subseteq \{T>mN\}$ for every $m$ and the bound tends to zero, ${\mathbb{P}}_i(T<\mathrm{\infty })=1$. From either boundary state $T=0$, so every start hits $A_N$ almost surely.

By [A1, F8] and step 3.1, the hypotheses of the bounded Dirichlet theorem hold for the chain with kernel $K_N$, boundary $A_N$, and data $f_N$. Its expected boundary payoff is the unique bounded solution of those equations. Step 1.2 shows that this solution is $g_N$.

For a path with $T<\mathrm{\infty }$, the endpoints are distinct and $T$ is the first visit to one of them, so $f_N(X_T)=1$ exactly when $T_N<T_0$. If $T=\mathrm{\infty }$, both hitting times are infinite and the theorem's payoff is zero, so the same indicator identity holds. Since $T<\mathrm{\infty }$ almost surely by step 3.1, [F8, step 4.1] yield ${\mathbb{P}}_i(T_N<T_0)={\mathbb{E}}_i[f_N(X_T)]=g_N(i)=\frac{i}{N}\mathrm{.}$

The assumption $N\ge 1$ makes $E_N$ nonempty and $0,N$ distinct, so an empty state space, a one-state space, or an empty boundary set is inapplicable. For $N=1$ both states are boundary states and there is no interior equation; for $N=2$ the single interior equation in step 1.2 applies. At $i=0$, the time-zero convention gives $T_0=0$ and both sides are zero; at $i=N$, it gives $T_N=0<T_0$ and both sides are one. All unlisted transition weights are zero, and the boundary rows are absorbing. AC is assumed and used through the canonical chain laws, conditional-expectation properties and bounded-future Markov identity, and the Dirichlet theorem. The claim is a hitting-probability identity, not an iff statement.