• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.739-751
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• 1995

Let ${X(t) : 0 \leq t < \infty}$ be an almost surely continuous Gaussian process with X(0) = 0, E{X(t)} = 0 and stationary increments $E{X(t) - X(s)}^2 = \sigma^2($\mid$t - s$\mid$)$, where $\sigma(y)$ is a function of $y \geq 0(e.g., if {X(t);0 \leq t < \infty}$ is a standard Wiener process, then $\sigma(t) = \sqrt{t})$. Assume that $\sigma(t), t > 0$, is a nondecreasing continuous, regularly varying function at infinity with exponent $\gamma$ for some $0 < \gamma < 1$.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.75-93
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• 2008

The rate of convergence of the distribution of order statistics to the corresponding extreme-value distribution may be characterized by the uniform and total variation metrics. de Haan and Resnick [4] derived the convergence rate when the second order generalized regularly varying function has second order derivatives. In this paper, based on the properties of the generalized regular variation and the second order generalized variation and characterized by uniform and total variation metrics, the convergence rates of the distribution of the largest order statistic are obtained under weaker conditions.

• East Asian mathematical journal
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• v.24 no.3
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• pp.251-261
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• 2008

Let {${\xi}_j;j\;{\geq}\;1$} be a centered strictly stationary random sequence defined by $S_0\;=\;0$, $S_n\;=\;\Sigma^n_{j=1}\;{\xi}_j$ and $\sigma(n)\;=\;33\sqrt {ES^2_n}$ where $\sigma(t),\;t\;>\;0$, is a nondecreasing continuous regularly varying function. Suppose that there exists $n_0\;{\geq}\;1$ such that, for any $n\;{\geq}\;n_0$ and $0\;{\leq}\;{\varepsilon}\;<\;1$, there exist positive constants $c_1$ and $c_2$ such that $c_1e^{-(1+{\varepsilon})x^2/2}\;{\leq}\;P\{\frac{{\mid}S_n{\mid}}{\sigma(n)}\;{\geq}\;x\}\;{\leq}\;c_2e^{-(1-{\varepsilon})x^2/2$, $x\;{\geq}\;1$ Under some additional conditions, we investigate some limsup results for the increments of partial sum processes of the sequence {${\xi}_j;j\;{\geq}\;1$}.

• Proceedings of the Korean Statistical Society Conference
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• 2000.11a
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• pp.115-118
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• 2000

In this paper we establish limit results on the increments of (N, d)-Gaussian processes with independent components, via estimating upper bounds of large deviation probabilities on the suprema of (N, d)-Gaussian processes.

• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.865-874
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• 1998

A modulus of continuity on the increments of a two-parameter Gaussian process is obtained via estimating large deviation probability inequalities on the suprema of the Gaussian process.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1265-1278
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• 2005

In this paper, we establish limit theorems containing both the moduli of continuity and the large incremental results for finite dimensional Gaussian processes with N parameters, via estimating upper bounds of large deviation probabilities on suprema of the Gaussian processes.

• Journal of the Korean Statistical Society
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• v.26 no.1
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• pp.57-74
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• 1997

Csorgo-Revesz type theorems for Wiener process are developed to those for Gaussian process. In particular, some results of superior and inferior limits for the increments of a Gaussian process are differently obtained under mild conditions, via estimating probability inequalities on the suprema of a Gaussian process.

• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1441-1451
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• 2007

A general law of the iterated logarithm (LIL) is established for $l^p$-valued Gaussian random fields under explicit conditions.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.813-822
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• 1996

Let ${W(t); 0 \leq t < \infty}$ be a standard Wiener process on a probability space $(\Omega, \Im , P)$.

• Journal of the Korean Mathematical Society
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• v.41 no.6
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• pp.1023-1034
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• 2004

In this paper we establish limsup and liminf theorems for the increments of partial sum processes of a dependent stationary Gaussian sequence.