• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.1
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• pp.9-16
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• 2005

The random signals defined as sums of the single frequency sinusoidal signals with random amplitudes and random phases or equivalently sums of functions obtained by adding a Sine and a Cosine function with random amplitudes, are used in the double randomization method for the Monte Carlo solution of the turbulent systems. We show that these random signals can be used for studying the properties of the Johnson noise by proving that constant multiples of these signals with uniformly distributed frequencies in a fixed frequency band satisfy the properties of the Gaussian white noise.

• Ocean Systems Engineering
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• v.7 no.3
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• pp.275-298
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• 2017

This paper provides a practical stochastic method by which the maximum equilibrium scour depth below a pipeline exposed to random waves plus a current on mild slopes can be derived. The approach is based on assuming the waves to be a stationary narrow-band random process, adopting the Battjes and Groenendijk (2000) wave height distribution for mild slopes including the effect of breaking waves, and using the empirical formulas for the scour depth on the horizontal seabed by Sumer and Fredsøe (1996). The present approach is valid for wave-dominant flow conditions. Results for random waves alone and random wave plus currents have been presented and discussed by varying the seabed slope and water depth. An approximate method is also proposed, and comparisons are made with the present stochastic method. For random waves alone it appears that the approximate method can replace the stochastic method, whereas the stochastic method is required for random waves plus currents. Tentative approaches to related random wave-induced scour cases for random waves alone are also suggested.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.291-301
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• 2010

This paper consists of two main parts. Firstly, we introduce an evolving binomial process from a binomial stock model and consider various types of limiting behavior of the logarithm of the evolving binomial process. Among others we find that the logarithm of the binomial process converges weakly to a Gaussian process. Secondly, we provide new approaches for proving the limit theorems for an integral process motivated by the evolving binomial process. We provide a new proof for the uniform strong LLN for the integral process. We also provide a simple proof of the functional CLT by using a restriction of Bernstein inequality and a restricted chaining argument. We apply the functional CLT to derive the LIL for the IID random variables from that for Gaussian.

• Ocean Systems Engineering
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• v.6 no.2
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• pp.161-189
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• 2016

This paper provides a practical stochastic method by which the maximum equilibrium scour depth around a vertical pile exposed to random waves plus a current on mild slopes can be derived. The approach is based on assuming the waves to be a stationary narrow-band random process, adopting the Battjes and Groenendijk (2000) wave height distribution for mild slopes including the effect of breaking waves, and using the empirical formulas for the scour depth on the horizontal seabed by Sumer and Fredsøe (2002). The present approach is valid for wave-dominant flow conditions. Results for random waves alone and random wave plus currents have been presented and discussed by varying the seabed slope and water depth. An approximate method is also proposed, and comparisons are made with the present stochastic method. For random waves alone it appears that the approximate method can replace the stochastic method, whereas the stochastic method is required for random waves plus currents. Tentative approaches to related random wave-induced scour cases on mild slopes are also suggested.

• Wind and Structures
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• v.38 no.1
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• pp.1-13
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• 2024

In order to effectively simulate nonstationary stochastic turbulent wind fields, four separation hybrid (SEP-H) models are proposed in the present study. Based on the assumption that the lateral turbulence component at one single-point is uncorrelated with the longitudinal and vertical turbulence components, the fluctuating wind is separated into 2nV-1D and nV1D nonstationary stochastic vector processes. The first process can be expressed as double proper orthogonal decomposition (DPOD) or proper orthogonal decomposition and spectral representation method (POD-SRM), and the second process can be expressed as POD or SRM. On this basis, four SEP-H models of nonstationary stochastic turbulent wind fields are developed. In addition, the orthogonal random variables in the SEP-H models are presented as random orthogonal functions of elementary random variables. Meanwhile, the number theoretical method (NTM) is conveniently adopted to select representative points set of the elementary random variables. The POD-FFT (Fast Fourier transform) technique is introduced in frequency to give full play to the computational efficiency of the SEP-H models. Finally, taking a long-span bridge as the engineering background, the SEP-H models are compared with the dimension-reduction DPOD (DR-DPOD) model to verify the effectiveness and superiority of the proposed models.

• Honam Mathematical Journal
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• v.9 no.1
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• pp.113-119
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• 1987

Random variables $X_1$, $X_1$, ..., $X_m$ are said to be weakly associated if whenever $\pi$ is a permutation of {1, 2,..., m}, $1{\leq}k<m$, and f: $R^{k}{\rightarrow}R$, g: $R^{m-k}{\rightarrow}R$ are coordinatewise nondecreasing functions then Cov $[f(X_{x(1)},...,\;X_{\pi(k)},\;g(X_{x(k+1)},...,\;X_{x(m)})]{\geq}0$, whenever the covariance is defined. An infinite collection of random variables is weakly associated if every finite subcollection is weakly associated. The basic properties of weak association and central limit theorem for weakly associated random variables are derived. We also extend this idea to point random fields and prove that a Cox process with a stationary weakly associated intensity rardom measure is weakly associated. Another inequalities and the fact that positive correlated normal random variables are weakly associated are also proved.

• Proceedings of the KSME Conference
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• 2004.04a
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• pp.508-512
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• 2004

An aerospace vehicle in flight can be exposed to random gust which may cause critical structural failure. In this paper, the failure analysis is conducted for composite wing subjected to random gust. For this, the profile of random gust is idealized as a stationary Gaussian random process and the power spectral density (PSD) of wing bending moment induced by gust is obtained. The PSD function is converted to probabilistic distributions and the failure probability during total flight time is calculated by Monte Carlo simulation.

• 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 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 the Korean Data and Information Science Society
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• v.17 no.1
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• pp.195-204
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• 2006

In this paper, we consider a renewal process in which both the inter-arrival times and rewards are fuzzy random variables. We prove the uniform levelwise convergence of fuzzy renewal and fuzzy renewal rewards. These results improve the result of Popova and Wu[European J. Oper. Research 117(1999), 606-617] and the main result of Hwang [Fuzzy Sets and Systems 116 (2000), 237-244].