Martingale theory: Collection of examples and counterexamples

The aim of this question is to collect interesting examples and counterexamples in martingale theory. There is a huge variety of such (counter)examples available here on StackExchange but I always have a hard time when I try to locate a specific example/question. I believe that it would be a benefit to make this knowledge easier to access. For this reason I would like to create a (big) list with references to related threads.

Martingale theory is a broad topic, and therefore I suggest to focus on time-discrete martingales $(M_n)_{n \in \mathbb{N}}$. I am well aware that this is still a quite broad field. To make this list a helpful tool (e.g. for answering questions) please make sure to give a short but concise description of each (counter)example which you list in your answer.

Related literature:

• Jordan M. Stoyanov: Counterexamples in Probability, Dover.
• Joseph P. Romano, Andrew F. Siegel: Counterexamples in probability and statistics, CRC Press.

convergence results:

• pointwise convergence of martingale $M_n$ does not imply $\sup_n \mathbb{E}(M_n^+)<\infty$ (this means that the converse of the martingal convergence theorem does not hold true)

• martingale which converges almost surely but not in $L^1$ (see also here)

• martingale $(M_n)_n$ such that $M_n \to -\infty$ almost surely (consider $-M_n$ to get a martingale such that $M_n \to \infty$ a.s.)

• non-trivial martingale which converges almost surely to $0$ (see also here)

• martingale which converges in probability but not almost surely (see also here)

• martingale which converges in distribution but not almost surely/in probability (see also Section 2.2 here)

uniform integrability:

• martingale $(M_n)_n$ for which $M_{\infty} = \lim_n M_n$ exists a.s. but $\mathbb{E}(M_{\infty} \mid \mathcal{F}_n) \neq M_n$ (see also here and here; note that such a martingale cannot be uniformly integrable and cannot converge in $L^1$)

• uniformly integrable martingale $(M_n)_n$ such that $\mathbb{E}\left( \sup_{n \in \mathbb{N}} |M_n| \right) = \infty$.

sample path behaviour:

• oscillating martingale with bounded sample paths

• non-trivial martingale which is constant with positive probability

• martingale which is non-constant and non-negative

Stopping times (Optional stopping/sampling theorem):

• martingale $(M_n)_{n \in \mathbb{N}}$ and stopping time $\tau$ such that $\mathbb{E}(M_{\tau}) \neq \mathbb{E}(M_0)$

• martingale $(M_n)_{n \in \mathbb{N}}$ and stopping time $\tau$ such that $M_{n \wedge \tau} \to M_{\tau}$ almost surely but not in $L^1$ (see the very first part of the the linked answer)

• martingale $(M_n)_{n \in \mathbb{N}}$ and stopping time $\tau$ such that $\tau<\infty$ almost surely and $\mathbb{E}(\tau)=\infty$

Other

• stochastic process $(M_n)_n$ which satisfies $\mathbb{E}(M_{n+1} \mid M_n) = M_n$ for all $n$ but which is not a martingale

• martingale which is not bounded in $L^1$