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ON THE RATES OF THE ALMOST SURE CONVERGENCE FOR SELF-NORMALIZED LAW OF THE ITERATED LOGARITHM
Pang, Tian-Xiao Korean Mathematical Society 2011 대한수학회보 Vol.48 No.6
Let {$X_i$, $i{\geq}1$} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote $S_n={\sum}_{i=1}^n\;X_i$, $M_n=max_{1{\leq}i{\leq}n}\;{\mid}S_i{\mid}$ and $V_n^2={\sum}_{i=1}^n\;X_i^2$. Then for d > -1, we showed that under some regularity conditions, $$\lim_{{\varepsilon}{\searrow}0}{\varepsilon}^2^{d+1}\sum_{n=1}^{\infty}\frac{(loglogn)^d}{nlogn}I\{M_n/V_n{\geq}\sqrt{2loglogn}({\varepsilon}+{\alpha}_n)\}=\frac{2}{\sqrt{\pi}(1+d)}{\Gamma}(d+3/2)\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2d+2}}\;a.s.$$ holds in this paper, where If g denotes the indicator function.
A SELF-NORMALIZED LIL FOR CONDITIONALLY TRIMMED SUMS AND CONDITIONALLY CENSORED SUMS
Pang Tian Xiao,Lin Zheng Yan Korean Mathematical Society 2006 대한수학회지 Vol.43 No.4
Let {$X,\;X_n;n\;{\geq}\;1$} be a sequence of ${\imath}.{\imath}.d.$ random variables which belong to the attraction of the normal law, and $X^{(1)}_n,...,X^{(n)}_n$ be an arrangement of $X_1,...,X_n$ in decreasing order of magnitude, i.e., $\|X^{(1)}_n\|{\geq}{\cdots}{\geq}\|X^{(n)}_n\|$. Suppose that {${\gamma}_n$} is a sequence of constants satisfying some mild conditions and d'($t_{nk}$) is an appropriate truncation level, where $n_k=[{\beta}^k]\;and\;{\beta}$ is any constant larger than one. Then we show that the conditionally trimmed sums obeys the self-normalized law of the iterated logarithm (LIL). Moreover, the self-normalized LIL for conditionally censored sums is also discussed.
PRECISE RATES IN THE LAW OF THE LOGARITHM FOR THE MOMENT CONVERGENCE OF I.I.D. RANDOM VARIABLES
Pang, Tian-Xiao,Lin, Zheng-Yan,Jiang, Ye,Hwang, Kyo-Shin Korean Mathematical Society 2008 대한수학회지 Vol.45 No.4
Let {$X,\;X_n;n{\geq}1$} be a sequence of i.i.d. random variables. Set $S_n=X_1+X_2+{\cdots}+X_n,\;M_n=\max_{k{\leq}n}|S_k|,\;n{\geq}1$. Then we obtain that for any -1<b<1/2, $\lim\limits_{{\varepsilon}{\searrow}0}\;{\varepsilon}^{2b+2}\sum\limits_{n=1}^\infty\;{\frac {(log\;n)^b}{n^{3/2}}\;E\{M_n-{\varepsilon}{\sigma}\sqrt{n\;log\;n\}+=\frac{2\sigma}{(b+1)(2b+3)}\;E|N|^{2b+3}\sum\limits_{k=0}^\infty\;{\frac{(-1)^k}{(2k+1)^{2b+3}$ if and only if EX=0 and $EX^2={\sigma}^2<{\infty}$.
A self-normalized LIL for conditionally trimmed sums and conditionally censored sums
Tian-xiao Pang,Zheng-yan Lin 대한수학회 2006 대한수학회지 Vol.43 No.4
Let {X, X_n; nge 1} be a sequence of i.i.d. randomvariables which belong to the attraction of the normal law, andX_n^{(1)}, ldots, X_n^{(n)} be an arrangement of X_1, ldots,X_n in decreasing order of magnitude, i.e., |X_n^{(1)}|gecdots ge |X_n^{(n)}|. Suppose that {r_n} is a sequence ofconstants satisfying some mild conditions and d^{'}(t_{n_k}) isan appropriate truncation level, where n_k=[beta^k] and betais any constant larger than one. Then we show that theconditionally trimmed sums obeys the self-normalized law of theiterated logarithm (LIL). Moreover, the self-normalized LIL for
Precise rates in the law of the logarithm for the moment convergence of i.i.d. random variables
Tian-Xiao Pang,Ye Jiang,황교신,Zheng-Yan Lin 대한수학회 2008 대한수학회지 Vol.45 No.4
Let {X,Xn; n ≥ 1} be a sequence of i.i.d. random variables. Set Sn = X₁ + X₂ + ... + Xn, Mn =[수식] Then we obtain that for any -1 < b < 1/2, [수식] if and only if EX = 0 and EX² = σ² < ∞. Let {X,Xn; n ≥ 1} be a sequence of i.i.d. random variables. Set Sn = X₁ + X₂ + ... + Xn, Mn =[수식] Then we obtain that for any -1 < b < 1/2, [수식] if and only if EX = 0 and EX² = σ² < ∞.
On the rates of the almost sure convergence for self-normalized law of the iterated logarithm
Tian-Xiao Pang 대한수학회 2011 대한수학회보 Vol.48 No.6
Let {X_i, i≥ 1} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote S_n=[수식]=max_(1≤ i≤ n)|S_i| and [수식]=[수식]. Then for d>-1, we showed that under some regularity conditions, [수식]=[수식]holds in this paper, where I{ · } denotes the indicator function.
Precise asymptotics in the law of the logarithm for the rescaled range statistic
Zheng-Yan Lin,Tian-Xiao Pang,황교신 한국통계학회 2013 Journal of the Korean Statistical Society Vol.42 No.2
For a sequence of i.i.d. zero mean random variables belonging to the domain of attraction of the normal law, two results concerning the rescaled range statistic are investigated in this paper. More specifically, we obtain precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence under some mild conditions.