• Honam Mathematical Journal
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• v.30 no.3
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• pp.479-486
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• 2008

Let {${\Omega}$, F, P} be a probability space and {$X_n{\mid}n{\geq}1$} be a sequence of random variables defined on it. We study the Hajeck-Renyi-type inequality for p..mixing random variable sequences and obtain the strong law of large numbers by using this inequality. We also consider the strong law of large numbers for weighted sums of ${\tilde{\rho}}$-mixing sequences.

• 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 $\sigma^2=EX_1^2+ 2\sum\limits{^{\infty}_{i=2}}E(X_1,X_i)=1$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1139-1153
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• 2010

Let {$X_j,\;j\geq1$} be a strictly stationary $\phi$-mixing sequence of non-degenerate random variables with $EX_1$ = 0. In this paper, we establish a self-normalized weak invariance principle and a central limit theorem for the sequence {$X_j$} under the condition that L(x) := $EX_1^2I{|X_1|{\leq}x}$ is a slowly varying function at $\infty$, without any higher moment conditions.

• Journal of the Korean Statistical Society
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• v.32 no.2
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• pp.203-211
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• 2003

A new result of weak convergence of the empirical process is established for a stationary ${\phi}-mixing$ sequence of random variables, which relaxes the existing conditions on mixing coefficients. The result is basically obtained from bounds for even moments of sums of ${\phi}-mixing$ r.v.'s useful for handling triangular arrays with entries decreasing in size.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1101-1111
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• 2008

Under the condition of h-integrability and appropriate conditions on the array of weights, we establish complete convergence and strong law of large numbers for weighted sums of an array of dependent random variables.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.201-210
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• 2011

In this paper, we obtain the H$\`{a}$jeck-R$\`{e}$nyi type inequality and the strong law of large numbers for asymptotically linear negative quadrant dependent random variables by using this inequality. We also give the strong law of large numbers for the linear process under asymptotically linear negative quadrant dependence assumption.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.401-408
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• 2009

Let {$Y_i,-{\infty}<i<{\infty}$} be a doubly infinite sequence of identically distributed and ${\rho}^*$-mixing random variables and {$a_i,-{\infty}<i<{\infty}$} an absolutely summable sequence of real numbers. In this paper, we prove the complete convergence of $\{\sum\limits_{k=1}^n\;\sum\limits_{n=-\infty}^\infty\;a_{i+k}Y_i/n^{1/t};\;n{\geq}1\}$ under suitable conditions.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.235-249
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• 2012

In the paper by Liu and Lin (Statist. Probab. Lett. 76 (2006), no. 16, 1787-1799), a new kind of precise asymptotics in the law of large numbers for the sequence of i.i.d. random variables, which includes complete convergence as a special case, was studied. This paper is devoted to the study of this new kind of precise asymptotics in the law of large numbers for moving average process under $\phi$-mixing assumption and some results of Liu and Lin [6] are extended to such moving average process.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.597-606
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• 2008

Let {$Y_{ij}-{\infty}\;<\;i\;<\;{\infty}$} be a doubly infinite sequence of identically distributed and ${\rho}^*$-mixing random variables with zero means and finite variances and {$a_{ij}-{\infty}\;<\;i\;<\;{\infty}$} an absolutely summable sequence of real numbers. In this paper, we prove the complete moment convergence of {${\sum}^n_{k=1}\;{\sum}^{\infty}_{i=-{\infty}}\;a_{i+k}Y_i/n^{1/p}$; $n\;{\geq}\;1$} under some suitable conditions. We extend Theorem 1.1 of Li and Zhang [Y. X. Li and L. X. Zhang, Complete moment convergence of moving average processes under dependence assumptions, Statist. Probab. Lett. 70 (2004), 191.197.] to the ${\rho}^*$-mixing case.

• Communications for Statistical Applications and Methods
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• v.21 no.3
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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.