Precise asymptotics in the deviation probability series of self-normalized sums
نویسندگان
چکیده
منابع مشابه
A Supplement to Precise Asymptotics in the Law of the Iterated Logarithm for Self-normalized Sums
Let X, X1, X2, . . . be i.i.d. random variables with zero means, variance one, and set Sn = ∑n i=1 Xi, n ≥ 1. Gut and Spǎtaru [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P(|Sn| ≥ ε √ 2n log log n) is replaced by E{|Sn|/√n− ε √ 2 log log n}+ in their results, the new one is call...
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Let an integer n ≥ 2 and a vector of independent, identically distributed random variables X = (X1, . . . , Xn) be given with P(X = 0) = 0 and define the self-normalized sum Zn = ( Pn i=1 Xi)/( Pn i=1 X 2 i ) . We derive a formula for EZ n which enables us to prove that EZ 2 n ≥ 1 and that EZ n = 1 if and only if the summands are symmetrically distributed. The formula moreover suggests nonparam...
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Qing-pei Zang Department of Mathematics, Huaiyin Teachers College, Huaian 223300, China Correspondence should be addressed to Qing-pei Zang, [email protected] Received 25 December 2008; Revised 30 March 2009; Accepted 18 June 2009 Recommended by Jewgeni Dshalalow Let {X,Xn;n ≥ 1} be a sequence of independent and identically distributed i.i.d. random variables and X is in the domain of attra...
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متن کاملDeviation Inequalities for Sums of Weakly Dependent Time Series
The aim of this paper is to extend the Bernstein inequality from the independent case to some weakly dependent ones. We consider a sample (X1, . . . , Xn) = (X (n) 1 , . . . , X (n) n ) of a stationary process (X (n) t ) with values in a metric space (X , d). Let F be the set of 1-Lipschitz functions from X to [−1/2,1/2]. We are interested by the deviation of the partial sum S( f ) = ∑n i=1 f (...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2011
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2010.10.005