• 대한수학회논문집
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• 제20권3호
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• pp.531-538
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• 2005

In this paper we derive the central limit theorem for ${\sum}_{i=1}^n\;a_{ni}\xi_i$, where ${a_{ni},\;1\;{\leq}\;i\;{\leq}\;n}$ is a triangular array of nonnegative numbers such that $sup_n{\sum}_{i=1}^n\;a_{ni}^2\;<\;{\infty},\;max_{1{\leq}i{\leq}n}a_{ni}{\rightarrow}0\;as\;n\;{\rightarrow}\;{\infty}\;and\;\xi'_i\;s$ are a linearly negative quadrant dependent sequence. We also apply this result to consider a central limit theorem for a partial sum of a generalized linear process $X_n\;=\;\sum_{j=-\infty}^\infty\;a_k+_j{\xi}_j$.

• 대한수학회지
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• 제43권3호
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• pp.529-538
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• 2006

In this paper we derive the central limit theorem for ${\sum}^n_{i=l}\;a_{ni}{\xi}_{i},\;where\;\{a_{ni},\;1\;{\le}\;i\;{\le}n\}$ is a triangular array of non-negative numbers such that $sup_n{\sum}^n_{i=l}\;a^2_{ni}\;<\;{\infty},\;max_{1{\le}i{\le}n\;a_{ni}{\to}\;0\;as\;n{\to}{\infty}\;and\;{\xi}'_{i}s$ are a linearly positive quadrant dependent sequence. We also apply this result to consider a central limit theorem for a partial sum of a generalized linear process of the form $X_n\;=\;{\sum}^{\infty}_{j=-{\infty}}a_{k+j}{\xi}_{j}$.

• 대한수학회논문집
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• 제26권1호
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• pp.151-161
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• 2011

Let {$X_n$, $n{\geq}1$} be a sequence of asymptotically almost negatively associated random variables and $S_n=\sum^n_{i=1}X_i$. In the paper, we get the precise results of H$\acute{a}$jek-R$\acute{e}$nyi type inequalities for the partial sums of asymptotically almost negatively associated sequence, which generalize and improve the results of Theorem 2.4-Theorem 2.6 in Ko et al. ([4]). In addition, the large deviation of $S_n$ for sequence of asymptotically almost negatively associated random variables is studied. At last, the Marcinkiewicz type strong law of large numbers is given.

• Communications for Statistical Applications and Methods
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• 제21권3호
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• pp.245-251
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• 2014

Two exponential inequalities are established for a wide class of general weakly dependent sequences of random variables, called ${\psi}$-weakly dependent process which unify weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The ${\psi}$-weakly dependent process includes, for examples, stationary ARMA processes, bilinear processes, and threshold autoregressive processes, and includes essentially all classes of weakly dependent stationary processes of interest in statistics under natural conditions on the process parameters. The two exponential inequalities are established on more general conditions than some existing ones, and are proven in simpler ways.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1573-1581
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• 2010

we investigate the n-fold convolution of the uniform distributions. First, we are concerned with the explicit distribution function of the partial sum ${\zeta}_n$ when the random variables are independent and has identically uniform distribution, next, we determine the n-fold convolution distribution of ${\zeta}_n$ when the identically distributed condition is not satisfied.

• 충청수학회지
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• 제26권3호
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• pp.615-623
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• 2013

In this paper we establish a central limit theorem for weighted sums of $Y_n={\sum_{i=1}^{n}}a_n,_iX_i$, where $\{a_{n,i},\;n{\in}N,\;1{\leq}i{\leq}n\}$ is an array of nonnegative numbers such that ${\sup}_{n{\geq}1}{\sum_{i=1}^{n}}a_{n,i}^2$ < ${\infty}$, ${\max}_{1{\leq}i{\leq}n}a_{n,i}{\rightarrow}0$ and $\{X_i,\;i{\in}N\}$ is a sequence of linear negatively quadrant dependent random variables with $EX_i=0$ and $EX_i^2$ < ${\infty}$. Using this result we will obtain a central limit theorem for partial sums of linear processes.

• Journal of the Korean Statistical Society
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• 제12권1호
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• pp.10-17
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• 1983

Let $X^{(n)}_j, j=1,2,\cdots,n, n=1,2,\cdots$ be a triangular array of random variables which arise naturally in a study of ferromagnetism in statistical mechanics. This paper presents weak convergence of random function $W_n(t)$, an appropriately normalized partial sum process based on $S^{(n)}_n = X^{(n)}_i+\cdot+X^{(n)}_n$. The limiting process W(t) is shown to be Gaussian when weak dependence exists. At the critical point where the change form weak to strong dependence takes place, W(t) turns out to be non-Gaussian. Our results are direct extensions of work by Ellis and Newmam (1978). An example is considered and the relation of these results to critical phenomena is briefly explained.

• Communications for Statistical Applications and Methods
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• 제6권2호
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• pp.585-594
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• 1999

Let {$X_{nj}$, 1$\leq$j$\leq$n,j$\geq$1} be a triangular array of random variables which are neither independent nor identically distributed. The almost sure convergences of randomly weighted partial sums of the form $$\sum_n^{j=1}$$ $W_{nj}$$X_{nj} are studied where {Wnj 1$\leq$j$\leq$n, j$\geq$1} is a triangular array of random weights. Application regarding the Efron bootstrap is also introduced.

• 대한수학회보
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• 제48권5호
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• pp.1041-1046
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• 2011

Let {X, $X_{i};\;i{\geq}1$} be a sequence of independent and identically distributed positive random variables. Denote $S_n= \sum\array\\_{i=1}^nX_i$ and $S\array\\_n^{(k)}=S_n-X_k$ for n ${\geq}$1, $1{\leq}k{\leq}n$. Under the assumption of the finiteness of the second moments, we derive a type of the law of the iterated logarithm for $S\array\\_n^{(k)}$ and the limit point set for its certain normalization.