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

Authors Liu, Weidong
Shao, Qi-Man View this author's profile
Wang, Qiying
Issue Date 2013
Source Bernoulli , v. 19, (3), August 2013, p. 1006-1027
Summary Let X-1, X-2,... be independent random variables with zero means and finite variances, and let S-n = Sigma(n)(i=1) X-i and V-n(2) = Sigma(n)(i=1) X-i(2). A Cramer type moderate deviation for the maximum of the self-normalized sums max(1 <= k <= n) S-k/V-n is obtained. In particular, for identically distributed X-1, X-2, ... , it is proved that P(max(1 <= k <= n) S-k >= xV(n))/(1 - Phi(x)) -> 2 uniformly for 0 < x <= o(n(1/6)) under the optimal finite third moment of X-1.
ISSN 1350-7265
Language English
Format Article
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