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

Weidong Liu; Qi Man Shao; Qiying Wang

doi:10.3150/12-BEJ415

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

Weidong Liu
, Qi Man Shao
, Qiying Wang

Department of Mathematics

Research output: Contribution to journal › Journal Article › peer-review

9 Citations (Scopus)

Abstract

Let X¹,X², . . . be independent random variables with zero means and finite variances, and let Sⁿ = σ_n ⁱ⁼¹ Xⁱ and V ₂ⁿ σ_n ⁱ⁼¹ X₂ ⁱ . A Cramér type moderate deviation for the maximum of the self-normalized sums max1≤k≤n Sk/Vn is obtained. In particular, for identically distributed X¹,X ², . . . , it is proved that P(max1≤k≤n S^k ≥ xVⁿ)/(1φ(x)) → 2 uniformly for 0 < × ≤ o(n1/6) under the optimal finite third moment of X1

Original language	English
Pages (from-to)	1006-1027
Number of pages	22
Journal	Bernoulli
Volume	19
Issue number	3
DOIs	https://doi.org/10.3150/12-BEJ415
Publication status	Published - Aug 2013

Keywords

Independent random variables
Maximum of self-normalized sums

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10.3150/12-BEJ415

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@article{f472f57be7904afb8f0d4d0487b38a6b,

title = "Self-normalized Cram{\'e}r type moderate deviations for the maximum of sums",

abstract = "Let X1,X2, . . . be independent random variables with zero means and finite variances, and let Sn = σn i=1 Xi and V 2n σn i=1 X2 i . A Cram{\'e}r type moderate deviation for the maximum of the self-normalized sums max1≤k≤n Sk/Vn is obtained. In particular, for identically distributed X1,X 2, . . . , it is proved that P(max1≤k≤n Sk ≥ xVn)/(1φ(x)) → 2 uniformly for 0 < × ≤ o(n1/6) under the optimal finite third moment of X1",

keywords = "Independent random variables, Maximum of self-normalized sums",

author = "Weidong Liu and Shao, \{Qi Man\} and Qiying Wang",

year = "2013",

month = aug,

doi = "10.3150/12-BEJ415",

language = "English",

volume = "19",

pages = "1006--1027",

journal = "Bernoulli",

issn = "1350-7265",

publisher = "Bernoulli Society for Mathematical Statistics and Probability",

number = "3",

}

TY - JOUR

T1 - Self-normalized Cramér type moderate deviations for the maximum of sums

AU - Liu, Weidong

AU - Shao, Qi Man

AU - Wang, Qiying

PY - 2013/8

Y1 - 2013/8

N2 - Let X1,X2, . . . be independent random variables with zero means and finite variances, and let Sn = σn i=1 Xi and V 2n σn i=1 X2 i . A Cramér type moderate deviation for the maximum of the self-normalized sums max1≤k≤n Sk/Vn is obtained. In particular, for identically distributed X1,X 2, . . . , it is proved that P(max1≤k≤n Sk ≥ xVn)/(1φ(x)) → 2 uniformly for 0 < × ≤ o(n1/6) under the optimal finite third moment of X1

AB - Let X1,X2, . . . be independent random variables with zero means and finite variances, and let Sn = σn i=1 Xi and V 2n σn i=1 X2 i . A Cramér type moderate deviation for the maximum of the self-normalized sums max1≤k≤n Sk/Vn is obtained. In particular, for identically distributed X1,X 2, . . . , it is proved that P(max1≤k≤n Sk ≥ xVn)/(1φ(x)) → 2 uniformly for 0 < × ≤ o(n1/6) under the optimal finite third moment of X1

KW - Independent random variables

KW - Maximum of self-normalized sums

UR - https://www.webofscience.com/wos/woscc/full-record/WOS:000322355200011

UR - https://openalex.org/W3099872972

U2 - 10.3150/12-BEJ415

DO - 10.3150/12-BEJ415

M3 - Journal Article

SN - 1350-7265

VL - 19

SP - 1006

EP - 1027

JO - Bernoulli

JF - Bernoulli

IS - 3

ER -

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

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Keywords

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