# Cramér's moderate deviations for martingales with applications

Let (ξ_i,ℱ_i)_i≥1 be a sequence of martingale differences. Set X_n=∑_i=1^n ξ_i and ⟨ X ⟩_n=∑_i=1^n 𝐄(ξ_i^2|ℱ_i-1). We prove Cramér's moderate deviation expansions for 𝐏(X_n/√(⟨ X⟩_n)≥ x) and 𝐏(X_n/√(𝐄X_n^2)≥ x) as n→∞. Our results extend the classical Cramér result to the cases of normalized martingales X_n/√(⟨ X⟩_n) and standardized martingales X_n/√(𝐄X_n^2), with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.

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