• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.305-314
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• 1999

The Hsu-Robbins-erd s theorem states that if {$X_m,n\geq1$} is a sequence of independent and identically distributed random variables, then ${EX_1}^2<\infty$ and $EX_1$=0 if and only if ${\sum_{n=1}}^\infty\;P($\mid${\sum_{k=1}}^nX_k$\mid$\geqn\in)<\infty$ for every $\in$ > 0. Under some auxiliary conditions, Sp taru (1994) extended this to the case where the $X_n$ are independent, but their distributions come from a finite set. Pruss (1996) proved Sp taru's result under weaker conditions, The purpose of this paper is to improve Pruss conditions.

• East Asian mathematical journal
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• v.35 no.1
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• pp.67-75
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• 2019

We investigate the averaging value of a random sampling of a Dirichlet series with some condition using Poisson distribution. Our result is the following: Let $L(s)={\sum}^{\infty}_{n=1}{\frac{a_n}{n^s}}$ be a Dirichlet series that converges absolutely for Re(s) > 1. If $X_t$ is an increasing random sampling with Poisson distribution and there exists a number $0<{\alpha}<{\frac{1}{2}}$ such that ${\sum}_{n{\leq}u}a_n{\ll}u^{\alpha}$, then we have $${\mathbb{E}}L(1/2+iX_t)=O(t^{\alpha}{\sqrt{{\log}t}})$$, for all sufficiently large t in ${\mathbb{R}}$. As a result, we get the behaviour of $L({\frac{1}{2}}+it)$ such that L is a Dirichlet L-function or a modular L-function, when t is sampled by the Poisson distribution.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.28 no.10C
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• pp.936-941
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• 2003

Density evolution was developed as a method for computing the capacity of low-density parity-check(LDPC) codes under the sum-product algorithm [1]. Based on the assumption that the passed messages on the belief propagation model can be approximated well by Gaussian random variables, a modified and simplified version of density evolution technique was introduced in [2]. Recently, the min-sum algorithm was applied to the density evolution of LDPC codes as an alternative decoding algorithm in [3]. Next question is how the min-sum algorithm is combined with a Gaussian approximation. In this paper, the capacity of various rate LDPC codes is obtained using the min-sum algorithm combined with the Gaussian approximation, which gives a simplest way of LDPC code analysis. Unlike the sum-product algorithm, the symmetry condition [4] is not maintained in the min-sum algorithm. Therefore, the variance as well as the mean of Gaussian distribution are recursively computed in this analysis. It is also shown that the min-sum threshold under a gaussian approximation is well matched to the simulation results.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1137-1146
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• 2011

Let {$X_i$, $i{\geq}1$} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote $S_n={\sum}_{i=1}^n\;X_i$, $M_n=max_{1{\leq}i{\leq}n}\;{\mid}S_i{\mid}$ and $V_n^2={\sum}_{i=1}^n\;X_i^2$. Then for d > -1, we showed that under some regularity conditions, $$\lim_{{\varepsilon}{\searrow}0}{\varepsilon}^2^{d+1}\sum_{n=1}^{\infty}\frac{(loglogn)^d}{nlogn}I\{M_n/V_n{\geq}\sqrt{2loglogn}({\varepsilon}+{\alpha}_n)\}=\frac{2}{\sqrt{\pi}(1+d)}{\Gamma}(d+3/2)\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2d+2}}\;a.s.$$ holds in this paper, where If g denotes the indicator function.

• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.429-440
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• 2006

In statistical computing, it is often for researchers to need the distribution of a weighted sum of noncentral chi-square variables. In this case, it is very limited to know its exact distribution. There are many works to contribute to this topic, e.g. Imhof (1961) and Solomon-Stephens (1977). Imhof's method gives good approximation to the true distribution, but it is not easy to apply even though we consider the development of computer technology Solomon-Stephens's three moment chi-square approximation is relatively easy and accurate to apply. However, they skipped many details, and their simulation is limited to a weighed sum of central chi-square random variables. This paper gives details on Solomon-Stephens's method. We also extend their simulation to the weighted sum of non-central chi-square distribution. We evaluated approximated powers for homogeneous test and compared them with the true powers. Solomon-Stephens's method shows very good approximation for the case.

• Communications for Statistical Applications and Methods
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• v.22 no.4
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• pp.377-387
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• 2015

The area under the ROC curve can be represented by both Mann-Whitney and Wilcoxon rank sum statistics. Consider an ROC surface and manifold equal to three dimensions or more. This paper finds that the volume under the ROC surface (VUS) and the hypervolume under the ROC manifold (HUM) could be derived as functions of both conditional Mann-Whitney statistics and conditional Wilcoxon rank sum statistics. The nullhypothesis equal to three distribution functions or more are identical can be tested using VUS and HUM statistics based on the asymptotic large sample theory of Wilcoxon rank sum statistics. Illustrative examples with three and four random samples show that two approaches give the same VUS and $HUM^4$. The equivalence of several distribution functions is also tested with VUS and $HUM^4$ in terms of conditional Wilcoxon rank sum statistics.

• Journal of the Korean Mathematical Society
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• v.43 no.3
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• pp.529-538
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• 2006

In this paper we derive the central limit theorem for ${\sum}^n_{i=l}\;a_{ni}{\xi}_{i},\;where\;\{a_{ni},\;1\;{\le}\;i\;{\le}n\}$ is a triangular array of non-negative numbers such that $sup_n{\sum}^n_{i=l}\;a^2_{ni}\;<\;{\infty},\;max_{1{\le}i{\le}n\;a_{ni}{\to}\;0\;as\;n{\to}{\infty}\;and\;{\xi}'_{i}s$ are a linearly positive quadrant dependent sequence. We also apply this result to consider a central limit theorem for a partial sum of a generalized linear process of the form $X_n\;=\;{\sum}^{\infty}_{j=-{\infty}}a_{k+j}{\xi}_{j}$.

• Communications for Statistical Applications and Methods
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• v.16 no.4
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• pp.723-730
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• 2009

Missing values in time series can be treated as unknown parameters and estimated by maximum likelihood or as random variables and predicted by the expectation of the unknown values given the data. The purpose of this study is to impute missing values which are regarded as the maximum likelihood estimator and random variable in incomplete data and to compare with two methods using ARMA model. For illustration, the Mumps data reported from the national capital region monthly over the years 2001 ${\sim}$ 2006 are used, and results from two methods are compared with using SSF(Sum of square for forecasting error).

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.829-842
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• 2009

Let {$X_{ni}|1\;{\le}\;i\;{\le}\;n$, $n\;{\ge}\;1$} be an array of rowwise negatively associated random variables and let $\alpha$ > 1/2, 0 < p < 2 ${\alpha}p\;{\ge}\;1$. In this paper we discuss $n^{{\alpha}p-2}h(n)$ max $_{1\;{\le}\;k{\le}n}\;|\;{\sum}^k_{i=1}\;X_{ni}|/n^{\alpha}\;{\to}\;0$ completely as $n\;{\to}\;{\infty}$ under not necessarily identically distributed with a suitable conditions and h(x) > 0 is a slowly varying function as $x\;{\to}\;{\infty}$. In addition, we obtained that $n^{{\alpha}p-2}h(n)$ max $_{1\;{\le}\;k{\le}n}\;|\;{\sum}^k_{i=1}\;X_{ni}|/n^{\alpha}\;{\to}\;0$ completely as $n\;{\to}\;{\infty}$ if and only if $E|X_{11}|^ph(|X_{11}|^{1/\alpha})\;<\;{\infty}$ and $EX_{11}\;=\;0$ under identically distributed case and some corollaries are obtained.

• 한국ITS학회:학술대회논문집
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• 2010.05a
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• pp.10-13
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• 2010