Precise asymptotics in the self-normalized law of the iterated logarithm
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 X, X1, X2, . . . be i.i.d. random variables with mean zero and positive, finite variance σ2, and set Sn = X1 + . . . + Xn, n ≥ 1. We prove that, if EX2I{|X| ≥ t} = o((log log t)−1) as t →∞, then for any a > −1 and b > −1, lim 2↗1/√1+a ( 1 √ 1+a − 2)b+1 ∞n=1 (log n) a(log log n)b n P { maxk≤n |Sk| ≤ √ σ2π2n 8 log log n(2 + an) } = 4 π ( 1 2(1+a)3/2 )b+1Γ(b + 1), whenever an = o(1/ log log n).
متن کاملSelf-normalized laws of the iterated logarithm
Stronger versions of laws of the iterated logarithm for self-normalized sums of i.i.d. random variables are proved.
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Let {X n ; n ≥ 1} be a strictly stationary negatively associated sequence which satisfies EX 1 = 0, V ar(X 1) < ∞. Set S n = n k=1 X k , n ≥ 1, σ 2 = EX 2 1 + 2 ∞ k=2 EX 1 X k. In this paper, we prove that, for b > −1, lim ε0 ε 2(b+1) ∞ n=1 (log log n) b n log n P{|S n | ≥ εσ n log log n} = 2 b+1 (b + 1) √ π Γ(b + 3 2) holds if EX 2 1 (1 + log log |X 1 |) b−1 < ∞. The result of Gut and Spˇataru...
متن کاملPrecise asymptotics in the law of logarithm under dependence assumptions
Keywords: Law of the logarithm Mixing sequences Precise asymptotics Strong approximation Stationary a b s t r a c t In a recent paper by Spătaru [Precise asymptotics for a series of T. a precise asymptotics in the law of the logarithm for sequence of i.i.d. random variables has been established. In this paper we show that there is an analogous result for strictly stationary ϕ-mixing sequence. T...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2008
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2007.09.054