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Asymptotic properties for the loglog laws under positive association

In: Mathematica Slovaca, vol. 62, no. 5
Xiao-Rong Yang - Ke-Ang Fu
Detaily:
Rok, strany: 2012, 979 - 994
Kľúčové slová:
the law of the iterated logarithm, Chung-type law of the iterated logarithm, positive association, moment convergence, tail probability
O článku:
Let $\{Xn: n≥1\}$ be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $Sn=∑k=1nXk$, $Mn=\maxk≤ n|Sk|$, $n≥1$. Suppose that $0<σ2=EX12+2∑k=2EX1Xk<∞$. In this paper, we prove that if $E |X1|2+δ<∞$ for some $δ\in(0,1]$, and $∑j=n+1 Cov(X1,Xj) =O(n)$ for some $α>1$, then for any $b>-1/2$ \begin{align*} \limε\searrow 0ε2b+1n=1 ((( log log n)b-1/2) / (n3/2 log n)) E\{Mn-σε \sqrt{2n log log n}\}+& =((2-1/2-bE|N|2(b+1)) / ((b+1)(2b+1))) ∑k=0(((-1)k) / ((2k+1)2(b+1)))& \end{align*} and \begin{align*} \limε\nearrow∞ε-2(b+1)n=1 ((( log log n)b) / (n3/2 log n)) E\{σε \sqrt{((π2n) / (8 log log n))}-Mn\}+& =\frac{Γ(b+1/2)}{\sqrt{2}(b+1)} ∑k=0(((-1)k) / ((2k+1)2b+2)),& \end{align*} where $x+ = \max\{x, 0\}$, $N$ is a standard normal random variable, and $Γ(·)$ is a Gamma function.
Ako citovať:
ISO 690:
Yang, X., Fu, K. 2012. Asymptotic properties for the loglog laws under positive association. In Mathematica Slovaca, vol. 62, no.5, pp. 979-994. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0059-0

APA:
Yang, X., Fu, K. (2012). Asymptotic properties for the loglog laws under positive association. Mathematica Slovaca, 62(5), 979-994. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0059-0
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