• Proceedings of the Korean Statistical Society Conference
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• 2000.11a
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• pp.119-121
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• 2000

A functional central limit theorem is obtained for a stationary multivariate linear process of the form $X_t=\sum\limits_{u=0}^\infty{A}_{u}Z_{t-u}$, where {$Z_t$} is a sequence of strictly stationary m-dimensional linearly positive quadrant dependent random vectors with $E Z_t = 0$ and $E{\parallel}Z_t{\parallel}^2 <{\infty}$ and {$A_u$} is a sequence of coefficient matrices with $\sum\limits_{u=0}^\infty{\parallel}A_u{\parallel}<{\infty}$ and $\sum\limits_{u=0}^\infty{A}_u{\neq}0_{m{\times}m}$. AMS 2000 subject classifications : 60F17, 60G10.

• Honam Mathematical Journal
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• v.30 no.4
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• pp.703-711
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• 2008

Let ${{\xi}_k,k{\in}{\mathbb{Z}}}$ be an associated H-valued random variables with $E{\xi}_k$ = 0, $E{\parallel}{\xi}_k{\parallel}$ < ${\infty}$ and $E{\parallel}{\xi}_k{\parallel}^2$ < ${\infty}$ and {$a_k,k{\in}{\mathbb{Z}}$} a sequence of bounded linear operators such that ${\sum}^{\infty}_{j=0}j{\parallel}a_j{\parallel}_{L(H)}$ < ${\infty}$. We define the sationary Hilbert space process $X_k={\sum}^{\infty}_{j=0}a_j{\xi}_{k-j}$ and prove that $n^{-1}{\sum}^n_{k=1}X_k$ converges to zero.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.1025-1034
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• 2010

Let {$X_n$, $n\;{\in}\;\mathbb{Z}_+^d$} be a field of identically distributed and negatively associated random variables with mean zero and set $S_n\;=\;{\sum}_{k{\leq}n}\;X_k$, $n\;{\in}\;\mathbb{Z}_+^d$, $d\;{\geq}\;2$. We investigate precise asymptotics for ${\sum}_n|n|^{r/p-2}P(|S_n|\;{\geq}\;{\epsilon}|n|^{1/p}$ and ${\sum}_n\;\frac{(\log\;|n|)^{\delta}}{|n|}P(|S_n|\;{\geq}\;{\epsilon}\;\sqrt{|n|\log|n|)}$, ($0\;{\leq}\;{\delta}\;{\leq}\;1$) as ${\epsilon}{\searrow}0$.

• Journal of Electrical Engineering and information Science
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• v.3 no.1
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• pp.8-13
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• 1998

Often ESOP(Exclusive Sum of Products) expressions provide more compact representations of logic functions and implemented circuits are known to be highly testable. Motivated by the merits of using XOR(Exclusive-OR) gates in circuit design, ESOP(Exclusive Sum of Products) expressions are considered s the input to the logic synthesis for random pattern testability. The problem of interest in this paper is whether ESOP expressions provide better random testability than corresponding SOP expressions of the given function. Since XOR gates are used to collect product terms of ESOP expression, fault propagation is not affected by any other product terms in the ESOP expression. Therefore the test set for a fault in ESOP expressions becomes larger than that of SOP expressions, thereby providing better random testability. Experimental results show that in many cases, ESOP expressions require much less random patterns compared to SOP expressions.

• Honam Mathematical Journal
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• v.32 no.3
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• pp.525-536
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• 2010

Let $X_1,X_2,\cdots$ be identically distributed $\rho$-mixing random variables with mean zeros and positive finite variances. In this paper, we prove $$\array{\lim\\{\in}\downarrow0}{\in}^2 \sum\limits_{n=3}^\infty\frac{1}{nlogn}P({\mid}S_n\mid\geq\in\sqrt{nloglogn}=1$$, $$\array{\lim\\{\in}\downarrow0}{\in}^2 \sum\limits_{n=3}^\infty\frac{1}{nlogn}P(M_n\geq\in\sqrt{nloglogn}=2 \sum\limits_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}$$ where $S_n=X_1+\cdots+X_n,\;M_n=max_{1{\leq}k{\leq}n}{\mid}S_k{\mid}$ and $\sigma^2=EX_1^2+ 2\sum\limits{^{\infty}_{i=2}}E(X_1,X_i)=1$.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.977-986
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• 2010

Let $X_1$, $X_2$, $\cdots$ be identically distributed negatively associated random variables with $EX_1\;=\;0$ and $E|X_1|^3$ < $\infty$. In this paper we prove $lim_{{\epsilon\downarrow}0}\;\frac{1}{-\log\;\epsilon}\sum\limits_{n=1}^\infty\frac{1}{n^2}ES_n^2I\{|S_n|\;{\geq}\;{\sigma\epsilon}n\}\;=\;2$ and $lim_{\epsilon\downarrow0}\;\epsilon^{2-p}\sum\limits_{n=1}^\infty\frac{1}{n^p}$ $E|S_n|^pI\{|S_n|\;{\geq}\;{\sigma\epsilon}n\}\;=\;\frac{2}{2-p}$ for 0 < p < 2, where $S_n\;=\;\sum\limits_{i=1}^{n}X_i$ and 0 < $\sigma^2\;=\;EX_1^2\;+\;\sum\limits_{i=2}^{\infty}Cov(X_1,\;X_i)$ < $\infty$. We consider some results of i.i.d. random variables obtained by Liu and Lin(2006) under negative association assumption.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.263-275
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• 2010

Let {$X_n;n\;\geq\;1$} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n\;=\;{\sum}^n_{k=1}X_k$, $M_n\;=\;max_{k{\leq}n}|S_k|$, $n\;{\geq}\;1$. Suppose $\sigma^2\;=\;EX^2_1+2{\sum}^\infty_{k=2}EX_1X_k$ (0 < $\sigma$ < $\infty$). We prove that for any b > -1/2, if $E|X|^{2+\delta}$(0<$\delta$$\leq$1), then $$lim\limits_{\varepsilon\searrow0}\varepsilon^{2b+1}\sum^{\infty}_{n=1}\frac{(loglogn)^{b-1/2}}{n^{3/2}logn}E\{M_n-\sigma\varepsilon\sqrt{2nloglogn}\}_+=\frac{2^{-1/2-b}{\sigma}E|N|^{2(b+1)}}{(b+1)(2b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2(b+1)}}$$ and for any b > -1/2, $$lim\limits_{\varepsilon\nearrow\infty}\varepsilon^{-2(b+1)}\sum^{\infty}_{n=1}\frac{(loglogn)^b}{n^{3/2}logn}E\{\sigma\varepsilon\sqrt{\frac{\pi^2n}{8loglogn}}-M_n\}_+=\frac{\Gamma(b+1/2)}{\sqrt{2}(b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2b+2'}}$$, where $\Gamma(\cdot)$ is the Gamma function and N stands for the standard normal random variable.

• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.141-149
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• 1995

Let N denote the set of positive integers. Fix $d_1, d_2 \in N with d = d_1 + d_2$. Let X and Y be real random variables and let ${X_i : i \in N^d_1} and {Y_j : j \in N^d_2}$ be independent families of independent identically distributed random variables with $L(X) = L(X_i) and L(Y) = L(Y_j)$, where $L(\cdot)$ denote the law of $\cdot$.

• Communications for Statistical Applications and Methods
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• v.12 no.1
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• pp.117-123
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• 2005

A functional central limit theorem is obtained for a stationary linear process of the form $X_{t}= \sum\limits_{j=0}^\infty{a_{j}x_{t-j}}$, where {x_t} is a strictly stationary sequence of negatively associated random variables with suitable conditions and {a_j} is a sequence of real numbers with $\sum\limits_{j=0}^\infty|a_{j}|<\infty$.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.157-163
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• 2014

Let {$X_i$, $1{\leq}i{\leq}n$} be a sequence of i.i.d. sequence of positive random variables with common absolutely continuous cumulative distribution function F(x) and probability density function f(x) and $E(X^2)$ < ${\infty}$. The random variables X + Y and $\frac{(X-Y)^2}{(X+Y)^2}$ are independent if and only if X and Y have gamma distributions. In addition, the random variables $S_n$ and $\frac{\sum_{i=1}^{m}(X_i)^2}{(S_n)^2}$ with $S_n=\sum_{i=1}^{n}X_i$ are independent for $1{\leq}m$ < n if and only if $X_i$ has gamma distribution for $i=1,{\cdots},n$.