Cramér Type Moderate deviations for the Maximum of Self-normalized Sums

Abstract

Let {X , X i , i ≥ 1} be i.i.d. random variables, S k be the partial sum andIn this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that

Self-normalized Cramér type moderate deviations for martingales

“…where O(1) is bounded by an absolute constant. The expansion (1.3) was further extended to independent but not necessarily identically distributed random variables by [Jing, Shao and Wang, 2003] For further self-normalized Cramér type moderate deviation results for independent random variables we refer, for example, to [Hu, Shao and Wang, 2009], [Liu, Shao and Wang, 2013], and [Shao and Zhou, 2016]. We also refer to [de la Peña, Lai and Shao, 2009] and for recent developments in this area.…”

Self-normalized Cramér type moderate deviations for the maximum of sums

“…For more related results, we refer to de la Peña, Lai and Shao [5] for a systematic treatment of the theory and applications of selfnormalization and Wang [13] for some refined self-normalized moderate deviations. As for the Cramér type moderate deviations for the maximum of self-normalized sums, namely for max 1≤k≤n S k /V n , Hu, Shao and Wang [7] were the first to prove that if X 1 , X 2 , . .…”

Estimating the Penalty Level of $\ell_{1}$-minimization via Two Gaussian Approximation Methods

“…This type of moderate deviation theorem was also applied in (Jia et al, 2019;Liu et al, 2013). Other type moderate deviation theorems can refer to (Hu et al, 2009;Jing et al, 2008;Liu and Shao, 2010;Shao and Zhou, 2016).…”