• Communications for Statistical Applications and Methods
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• v.16 no.4
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• pp.687-696
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• 2009

We prove a central limit theorem for the negatively associated random variables in a Hilbert space and extend this result to the linear process generated by negatively associated random variables in a Hilbert space. Our result implies an extension of the central limit theorem for the linear process in a real space under negative association to a simplest case of infinite dimensional Hilbert space.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.91-99
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• 2010

In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process generated by asymptotically almost negatively random variables in a Hilbert space with this result.

• Honam Mathematical Journal
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• v.36 no.3
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• pp.679-688
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• 2014

The notion of asymptotically negative dependence for collection of random variables is generalized to a Hilbert space and the almost sure convergence for these H-valued random variables is obtained. The result is also applied to a linear process generated by H-valued asymptotically negatively dependent random variables.

• Communications of the Korean Mathematical Society
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• v.23 no.1
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• pp.133-140
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• 2008

Let {${\xi}_k,\;k\;{\in}\;{\mathbb{Z}}$} be a strictly stationary associated sequence of H-valued random variables with $E{\xi}_k\;=\;0$ and $E{\parallel}{\xi}_k{\parallel}^2\;<\;{\infty}$ and {$a_k,\;k\;{\in}\;{\mathbb{Z}}$} a sequence of linear operators such that ${\sum}_{j=-{\infty}}^{\infty}\;{\parallel}a_j{\parallel}_{L(H)}\;<\;{\infty}$. For a linear process $X_k\;=\;{\sum}_{j=-{\infty}}^{\infty}\;a_j{\xi}_{k-j}$ we derive that {$X_k} fulfills the functional central limit theorem.

• 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.

• The Journal of the Acoustical Society of Korea
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• v.22 no.3E
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• pp.133-140
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• 2003

A time series simulation is presented by a ray approach for the simulating the received waveform of a broadband acoustical signals interacting with the ocean boundaries. The environment is assumed to be horizontally stratified, and the seafloor is described in terms of homogeneous fluid half-space. The ray approach includes the effects of reflection from the air-water, water-sediment interface and phase shifts due to boundaries interaction. To generate time series, we assume that the acoustic energy propagates from source to receiver along eigenrays and represent the action of the bottom on the incident wave by a linear filter and characterized in the frequency domain by the transfer function. As example application, the time series for an explosive source in a shallow water environment is calculated and analyzed in terms of acoustical process. good agreement with measured time series is demonstrated.