Cramér's moderate deviations for martingales with applications

Let (xi i,Fi)i >= 1 i , F i ) i >= 1 be a sequence of martingale differences. Set X n = ni =1 xi i and X n = n i =1 E (xi i 2 |F i -1 ) . We prove Cramer's moderate deviation expansions for P (X n / root X n >= x) and P (X n / E X n 2 >= x) as n ->infinity . Our results extend the classical Cramer result to the cases of normalized martingales Xn/root Xn n / root X n and standardized martingales Xn/ n / E X n 2 , with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.