Cramér large deviation expansions for martingales under Bernstein’s condition

Abstract

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.

Non-uniform Berry–Esseen bounds for martingales with applications to statistical estimation

“…This result has an exponential decaying rate in x, compablack to the polynomial decaying rate in the nonuniform Berry-Esseen bounds of Haeusler and Joos (1988b) and Joos (1991). The bounds (10) and (12) are closely related to the results of Fan et al (2013) and Grama and Haeusler (2000). However, we complete on these results in three aspects.…”

Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

“…The proof of Lemma 4 is complicated, and it is a refinement of the proof of Lemma 3.1 in [11]. Thus we give details in the supplemental article [13].…”

Self-normalized Cramér type moderate deviations for stationary sequences and applications

“…Following the seminal work of Cramér, various moderate deviation expansions for standardized sums have been obtained by many authors, see, for instance, Petrov [28], Saulis and Statulevičius [36] and [15]. See also Račkauskas [29,30], Grama [19], Grama and Haeusler [20] and [14] for martingales, and Wu and Zhao [38] and Cuny and Merlevède [9] for stationary processes. For establishing moderate deviation expansions of type (1.2) with a range 0 ≤ x = o(n α ), α > 0, Linnik's condition is necessary.…”