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A supermartingale is an integrable adapted process satisfying $\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]\le V_n$ almost surely. Its conditional expectation can decrease, although a realized sample path need not be decreasing. For a mean-zero random walk $S_n$, the process $V_n=S_n-cn$ with $c>0$ is a supermartingale: the conditional expected next change is $-c$. Negating a supermartingale gives a submartingale. A nonnegative supermartingale converges almost surely to a finite limit, but its limiting mean may be smaller than its earlier means without uniform integrability.

A time-changed filtration records the information available after a new clock has reached an old time. Let $({\mathcal{F}}_t)$ be a filtration and let $({\tau }_s)$ be a nondecreasing family of finite stopping times. DefineThe second expression is the stopping-time sigma-algebra. If $s\le u$ and $A\in {\mathcal{F}}_{{\tau }_s}$, thenThus ${\mathcal{G}}_s\subseteq {\mathcal{G}}_u$, as required for a filtration.

For the deterministic clock ${\tau }_s=2s$, the new information is ${\mathcal{F}}_{2s}$, not ${\mathcal{F}}_s$. A continuous adapted process sampled at these stopping times is adapted to the new filtration. Preservation of the local martingale property requires further hypotheses such as those in the optional time-change theorem; merely substituting arbitrary random times does not establish it.