Large deviations for martingales via Cramér's method

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

“…If the martingale differences are bounded |ξ i | ≤ ε n and satisfy condition (A2), Grama and Haeusler [17] proved the asymptotic expansion (2.5) for all x ∈ [0, α min{ε −1/2 n , δ −1 n }]. Now Theorem 1 holds for a larger range x ∈ [0, αε −1 n ] and a much more general class of martingales.…”

“…In this paper we are concerned with Cramér moderate deviations for martingales. When the martingale differences are bounded, we refer to Bose [3,4], Račkauskas [22,23,24], Grama and Haeusler [17]. Let (η i , F i ) i=0,...,n be a sequence of square integrable martingale differences defined on a probability space (Ω, F , P), where η 0 = 0 and {∅, Ω} = F 0 ⊆ ... ⊆ F n ⊆ F .…”

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.…”

Cramér type moderate deviations for self-normalized ψ-mixing sequences

“…Following the seminal work of Cramér, various moderate deviation expansions for standardized sums have been obtained by many authors (see, for instance, Petrov, 1954Petrov, , 1975Linnik, 1961;Saulis and Statulevičius, 1978;Fan, 2017). See also Račkauskas (1990Račkauskas ( , 1995, Grama (1997), Grama and Haeusler (2000), Fan et al (2013) for martingales.…”