Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations
نویسندگان
چکیده
منابع مشابه
Precise Asymptotics in Chung’s law of the iterated logarithm∗
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 کاملPrecise Asymptotics in the Law of the Iterated Logarithm under Dependence
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...
متن کاملA 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...
متن کاملOn the law of the iterated logarithm.
The law of the iterated logarithm provides a family of bounds all of the same order such that with probability one only finitely many partial sums of a sequence of independent and identically distributed random variables exceed some members of the family, while for others infinitely many do so. In the former case, the total number of such excesses has therefore a proper probability distribution...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematical Problems in Engineering
سال: 2021
ISSN: 1563-5147,1024-123X
DOI: 10.1155/2021/6691857