• 대한수학회지
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• 제49권5호
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• pp.1053-1064
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• 2012

For a double array of random elements $\{X_{mn};m{\geq}1,n{\geq}1\}$ in a $p$-uniformly smooth Banach space, $\{b_{mn};m{\geq}1,n{\geq}1\}$ is an array of positive numbers, convergence of double random series ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}X_{mn}$, ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}b^{-1}_{mn}X_{mn}$ and strong law of large numbers $$b^{-1}_{mn}\sum^m_{i=1}\sum^n_{j=1}X_{ij}{\rightarrow}0$$ as $$m{\wedge}n{\rightarrow}{\infty}$$ are established.

• Korean Journal of Mathematics
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• 제18권4호
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• pp.395-407
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• 2010

In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations $$cf\(\sum_{i=1}^{n}x_i\)+\sum_{j=2}^{n}f\(\sum_{i=1}^{n}x_i-(n+c-1)x_j\)\\=(n+c-1)\(f(x_1)+c\sum_{i=2}^{n}f(x_i)+\sum_{i<j,j=3}^{n}\(\sum_{i=2}^{n-1}f(x_i-x_j\)\),\\Q\(\sum_{i=1}^{n}d_ix_i\)+\sum_{1{\leq}i<j{\leq}n}d_id_jQ(x_i-x_j)=\(\sum_{i=1}^{n}d_i\)\(\sum_{i=1}^{n}d_iQ(x_i)\)$$ in random normed spaces.

• Nuclear Engineering and Technology
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• 제54권6호
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• pp.2084-2093
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• 2022

Probabilistic safety assessment is widely used to quantify the risks of nuclear power plants and their uncertainties. When the lognormal distribution describes the uncertainties of basic events, the uncertainty of the top event in a fault tree is approximated with the sum of lognormal random variables after minimal cutsets are obtained, and rare-event approximation is applied. As handling complicated analytic expressions for the sum of lognormal random variables is challenging, several approximation methods, especially Monte Carlo simulation, are widely used in practice for uncertainty analysis. In this study, a theoretical approach for analyzing the sum of lognormal random variables using an efficient numerical integration method is proposed for uncertainty analysis in probability safety assessments. The change of variables from correlated random variables with a complicated region of integration to independent random variables with a unit hypercube region of integration is applied to obtain an efficient numerical integration. The theoretical advantages of the proposed method over other approximation methods are shown through a benchmark problem. The proposed method provides an accurate and efficient approach to calculate the uncertainty of the top event in probabilistic safety assessment when the uncertainties of basic events are described with lognormal random variables.

• Communications for Statistical Applications and Methods
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• 제26권3호
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• pp.261-272
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• 2019

In this paper, we propose a procedure to build a prediction interval of the sum of dependent binary random variables over a graph to account for the dependence among binary variables. Our main interest is to find a prediction interval of the weighted sum of dependent binary random variables indexed by a graph. This problem is motivated by the prediction problem of various elections including Korean National Assembly and US presidential election. Traditional and popular approaches to construct the prediction interval of the seats won by major parties are normal approximation by the CLT and Monte Carlo method by generating many independent Bernoulli random variables assuming that those binary random variables are independent and the success probabilities are known constants. However, in practice, the survey results (also the exit polls) on the election are random and hardly independent to each other. They are more often spatially correlated random variables. To take this into account, we suggest a spatial auto-regressive (AR) model for the surveyed success probabilities, and propose a residual based bootstrap procedure to construct the prediction interval of the sum of the binary outcomes. Finally, we apply the procedure to building the prediction intervals of the number of legislative seats won by each party from the exit poll data in the $19^{th}$ and $20^{th}$ Korea National Assembly elections.

• 대한수학회지
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• 제16권2호
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• pp.107-111
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• 1980

• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.647-653
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• 2012

In this paper, we consider fuzzy random sets as (measurable) mappings from a probability space into the set of fuzzy sets and prove a uniform strong law of large numbers for sequences of independent and identically distributed fuzzy random sets. Our results generalize those of Bass and Pyke(1984)and Jang and Kwon(1998).

• 한국콘텐츠학회논문지
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• 제6권5호
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• pp.29-34
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• 2006

본 연구에서는, Banach 공간의 값을 갖는 확률요소들의 합 $S_n=\sum_{i=1}^nV-i$ 수렴하는 경우에, Tail 합 $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}V-i$에 대한 대수의 법칙을 고찰하여 $S_n$이 하나의 확률변수 S로 수렴하는 속도를 연구한다. 좀 더 구체적으로 말하자면, 확률변수들의 Tail 합과 확률요소들의 Tail 합에 대한 극한 성질의 유사성을 연구하여, Banach 공간에서 독립인 확률요소들의 Tail 합에 대한 약 대수의 법칙과 하나의 수렴법칙이 동등함을 기술하는 기존의 정리를 다른 대체적인 방법으로 증명한다.

• 호남수학학술지
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• 제30권1호
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• pp.9-20
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• 2008

Let ${X_i{\mid}i{\geq}1}$ be a strictly stationary sequence of associated random variables with mean zero and let ${\sigma}^2=EX_1^2+2\sum\limits_{j=2}^\infty{EX_1}{X_j}$ with 0 < ${\sigma}^2$ < ${\infty}$. Set $S_n={\sum\limits^n_{i=1}^\{X_i}$, the precise asymptotics for ${\varepsilon}^{{\frac{2(r-p)}{2-p}}-1}\sum\limits_{n{\geq}1}n^{{\frac{r}{p}}-{\frac{1}{p}}+{\frac{1}{2}}}P({\mid}S_n{\mid}{\geq}{\varepsilon}n^{{\frac{1}{p}}})$,${\varepsilon}^2\sum\limits_{n{\geq}3}{\frac{1}{nlogn}}p({\mid}Sn{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ and ${\varepsilon}^{2{\delta}+2}\sum\limits_{n{\geq}1}{\frac{(loglogn)^{\delta}}{nlogn}}p({\mid}S_n{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ as ${\varepsilon}{\searrow}0$ are established under the suitable conditions.

• 대한수학회논문집
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• 제25권1호
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• pp.119-128
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• 2010

Let ${X_n,\;n\geq1}$ be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space ($\Omega$,A,P), and let ${N_n,\;n\geq1}$ be a sequence of positive integer-valued r.vs., defined on the same probability space ($\Omega$,A,P). Furthermore, we assume that the r.vs. $N_n$, $n\geq1$ are independent of all r.vs. $X_n$, $n\geq1$. In present paper we are interested in asymptotic behaviors of the random sum $S_{N_n}=X_1+X_2+\cdots+X_{N_n}$, $S_0=0$, where the r.vs. $N_n$, $n\geq1$ obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums $S_{N_n}$ have been investigated in the theory probability and stochastic processes for quite some time (see [1], [4], [2], [3], [5]). Recently, the random sum approach is used in some applied problems of stochastic processes, stochastic modeling, random walk, queue theory, theory of network or theory of estimation (see [10], [12]). The main aim of this paper is to establish some results related to the asymptotic behaviors of the random sum $S_{N_n}$, in cases when the $N_n$, $n\geq1$ are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.

• 한국경영과학회지
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• 제4권1호
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• pp.69-77
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• 1979

A single machine stochastic scheduling problem is considered. Associated with each job is its random processing time and random deferral cost. The criterion is to order the jobs so as to minimize the sum of the deferral costs. The expected sum of the deferral costs is theroretically derived under the stochastic situation for each of several scheduling decision rules which are well known for the deterministic environment. It is also shown that certain stochastic problems can be reduced to equivalent deterministic problems. Two examples are illustrated to show the expected total deferral costs.