Moderate deviations for martingale differences and applications to φ -mixing sequences

Moderate Deviations for Recursive Stochastic Algorithms

“…Guillin [16] considers inhomogeneous functionals of a "fast" continuous time ergodic Markov chain, and in [17] this is extended to a small noise diffusion whose coefficients depend on the "fast" Markov chain. There are also results for martingale differences such as Dembo [7], Gao [14], and Djellout [9]. For various reasons, the issues previously mentioned regarding the difficulties in the proof of the upper bound and the simplification in the lower bound for degenerate noise do not play a role in these papers.…”

Moderate Deviations for I.I.D. Random Variables

“…These and similar techniques have been employed extensively by Ledoux [9] and Djellout [6]. We will comment on Ledoux's result and techniques a bit more extensively in Remark 2.8.…”

“…We are grateful to the Mathematisches Forschungsinstitut Oberwolfach for its kind hospitality. We are indebted to an unknown referee for bringing the references [9] and [6] to our attention.…”

Moderate deviations for the Durbin–Watson statistic related to the first-order autoregressive process

“…We are inspired by the recent paper of Bercu and Proïa [2], where the almost sure convergence and the central limit theorem are established for both the least squares estimators and the Durbin-Watson statistic. Our results are proved via an extensive use of the results of Dembo [5], Dembo and Zeitouni [6] and Worms [22], [23], [24] on the one hand, and of the paper of Puhalskii [19] and Djellout [7] on the other hand, about moderate deviations for martingales. In order to introduce the Durbin-Watson statistic, we shall focus our attention on the first-order autoregressive process given, for all n ≥ 1, by X n = θX n−1 + ε n ε n = ρε n−1 + V n (1.1) be arbitrarily chosen.…”