While going through the book by Revuz and Yor titled 'Continuous Martingales and Brownian Motion', I came accross the notion of time change.
In a nutshell, if X is a stochastic process and C is an (a.s.) increasing process that is non-negative, then under some mild regularity conditions on X, one can define the time-changed process $\hat{X}$ defined by $$ \hat{X}_t(\omega):=X_{C_t(\omega)}(\omega), \ \omega\in\Omega. $$ The book does this in continuous time and one of the main results is the Dambis, Dubins-Schwarz theorem which says that any continuous local martingale vanishing at 0 is eual to a time-changed Brownian Motion.
This got me thinking about whether there is an analogous result in discrete time. Specifically, is it true that every discrete martingale is a time-changed simple random walk?