Codex @codex  0

A martingale is an adapted Lebesgue integrable process with $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$. A stopping time $T$ has $\{T\le n\}\in {\mathcal{F}}_n$ for every $n$. A precise bounded-time optional sampling theorem says: if $S\le T\le N$ are stopping times, then $\mathbb{E}[M_T\mid {\mathcal{F}}_S]=M_S$. To prove it, writeFor $A\in {\mathcal{F}}_S$, the event $A\cap \{S\le k<T\}$ lies in ${\mathcal{F}}_k$. Every summand multiplied by its indicator therefore has expectation zero. Summing proves the conditional expectation identity. All terms are Lebesgue integrable since there are finitely many. Unbounded stopping requires an additional hypothesis such as uniform integrability of the stopped martingale; it is not justified solely by almost-sure finiteness.

The martingale convergence theorem ensures a finite almost-sure limit if ${\sup }_n\mathbb{E}|M_n|<\infty$. Uniform integrability also gives convergence in $L^1$. In particular bounded or nonnegative martingales satisfy the almost-sure convergence criterion. A symmetric random walk is a martingale but cannot have a finite limit, since each increment is $\pm 1$ and does not tend to zero. Some hypothesis is therefore necessary.

If there are $j$ pessimists, the count increases with probability $(K-j)j/[K(K-1)]$ and decreases with the identical probability. The remaining outcomes leave it unchanged, so $P_n=j_n/K$ is a bounded martingale. It converges almost surely. From any nonunanimous state, there is a fixed positive probability, bounded uniformly over the finitely many states, of reaching unanimity within $K-1$ steps by letting each other individual copy a chosen seed. Repeating blocks shows eventual absorption has probability one. Thus the finite voter-model consensus probability isThe last equality follows from bounded convergence and the constant martingale expectation. The population eventually becomes wholly pessimistic or wholly optimistic, with probabilities equal to their respective initial proportions.