Optional Stopping Theorem for Stochastic Processes with Constant Mean (and not a Martingale)

Question

Let $(X_n)_{n\geq 1}$ be a martingale with respect to $(Y_n)_{n\geq 1}$, i.e., the martingale condition $$ \mathbb{E}[X_n|Y_1, \ldots, Y_{n-1}] = X_{n-1} $$ holds. From this condition, it follows that the process $(X_n)$ has constant mean $\mathbb{E}[X_n] = \mathbb{E}[X_1]$ for all $n$. Under some assumptions, we then also have $\mathbb{E}[X_T] = \mathbb{E}[X_1]$, where $T$ is a stopping time. My question is, does this optional stopping theorem also apply to processes that have constant mean, but do not fulfill the martingale condition above? An example for such a process was given here: Example of a sequence of r.v.'s with constant stopping time that is not a Martingale

Answer 1

No, it does not hold. In fact, an equivalent definition of a martingale is an integrable adapted process that satisfies the optional stopping theorem.

As a counter-example, let your process be $X_n$ where $\mathbb{P}[X_i = 1] = 1/2$ and $\mathbb{P}[X_i = -1] = 1/2$ and $X_i$ are i.i.d. Let $\tau=\inf\{n: X_n =1\}$, then $\mathbb{E}[X_{\tau}]=1$.

Note that I am NOT taking the process to be a random walk. It's just a process which at each time $n$ takes the value $-1$ or $1$.