Bartlett type identities are shown to exist for martingales. As applications, we give a cumulant-based proof of the martingale central limit theorem, and we give an algorithm for calculating approximate cumulants of the least squares estimator in the AR(1) process.
"Bartlett Type Identities for Martingales." Ann. Statist. 22 (1) 21 - 38, March, 1994. https://doi.org/10.1214/aos/1176325355