• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.467-476
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• 2007

A complete convergence theorem for arrays of rowwise independent random variables was proved by Sung, Volodin, and Hu [14]. In this paper, we extend this theorem to the Banach space without any geometric assumptions on the underlying Banach space. Our theorem also improves some known results from the literature.

• Journal of applied mathematics & informatics
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• v.5 no.1
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• pp.125-132
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• 1998

In this paper we obtain a generalization of Chung's strong law of large numbers to the case of independent fuzzy random variables.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.45-55
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• 2016

Let {$X_n,n{\geq}1$} be a sequence of negatively superadditive dependent random variables. In the paper, we study the strong law of large numbers for general weighted sums ${\frac{1}{g(n)}}{\sum_{i=1}^{n}}{\frac{X_i}{h(i)}}$ of negatively superadditive dependent random variables with non-identical distribution. Some sufficient conditions for the strong law of large numbers are provided. As applications, the Kolmogorov strong law of large numbers and Marcinkiewicz-Zygmund strong law of large numbers for negatively superadditive dependent random variables are obtained. Our results generalize the corresponding ones for independent random variables and negatively associated random variables.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.4
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• pp.501-510
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• 2008

The distribution of the sum of n independent random variables having exponential distributions with different parameters ${\beta}_i$ ($i=1,2,{\ldots},n$) is given in [2], [3], [4] and [6]. In [1], by using Laplace transform, Jasiulewicz and Kordecki generalized the results obtained by Sen and Balakrishnan in [6] and established a formula for the distribution of this sum without conditions on the parameters ${\beta}_i$. The aim of this note is to present a method to find the distribution of the sum of n independent exponentially distributed random variables with different parameters. Our method can also be used to handle the case when all ${\beta}_i$ are the same.

• The Journal of Natural Sciences
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• v.11 no.1
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• pp.15-17
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• 1999

Let {f(n)} be an increasing sequence such that f(n)>0 for each n and f(n)$\rightarrow$$\infty$. Let {X$_n$,n$\geq1$} be a sequence of pairwise independent random variables. In this paper we give sufficient conditions on {X$_n$,n$\geq1$} such that $sum_{i=1}^n$(X$_i$-EX$_i$)/f(n) converges to zero almost surely.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1419-1443
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• 2021

The asymmetric Laplace distribution arises as a limiting distribution of geometric summations of independent and identically distributed random variables with finite second moments. The main purpose of this paper is to study the weak limit theorems for geometric summations of independent (not necessarily identically distributed) random variables together with convergence rates to asymmetric Laplace distributions. Using Trotter-operator method, the orders of approximations of the distributions of geometric summations by the asymmetric Laplace distributions are established in term of the "large-𝒪" and "small-o" approximation estimates. The obtained results are extensions of some known ones.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.363-370
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• 1995

A famous result of Chow and Robbins [8] asserts that if ${X_n, n \geq 1}$ are independent and identically distributed (i.i.d.) random variables with $E$\mid$X_1$\mid$ = \infty$, then for each sequence of constants ${M_n, n \geq 1}$ either $$ (1) lim inf_{n\to\infty} $\mid$\frac{M_n}{\sum_{j=1}^{n}X_j}$\mid$ = 0 almost certainly (a.c.) $$ or $$ (2) lim sup_{n\to\infty}$\mid$\frac{M_n}{\sum_{j=1}^{n}X_j}$\mid$ = \infty a.c. $$ and thus $P{lim_{n\to\infty} \sum_{j=1}^{n}X_j/M_n = 1} = 0$. Note that both (1) and (2) may indeed prevail.

• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.765-773
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• 2004

In this paper we define the notion of asymptotically quadrant independent random field and derive the strong laws of large numbers for this random field.

• Communications for Statistical Applications and Methods
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• v.27 no.3
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• pp.285-299
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• 2020

In many cases, we are interested in identifying independence between variables. For continuous random variables, correlation coefficients are often used to describe the relationship between variables; however, correlation does not imply independence. For finite discrete random variables, we can use the Pearson chi-square test to find independency. For the mixed type of continuous and discrete random variables, we do not have a general type of independent test. In this study, we develop a independence test of a continuous random variable and a discrete random variable without assuming a specific distribution using kernel density estimation. We provide some statistical criteria to test independence under some special settings and apply the proposed independence test to Pima Indian diabetes data. Through simulations, we calculate false positive rates and true positive rates to compare the proposed test and Kolmogorov-Smirnov test.

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.509-522
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• 2008

The main aim of this paper is to present some results related to asymptotic behavior of distribution functions of random variables of chi-square type $X^2_N={\Sigma}^N_{i=1}\;X^2_i$ with degrees of freedom N, where N is a positive integer-valued random variable independent on all standard normally distributed random variables $X_i$. Two ways for computing the distribution functions of chi-square type random variables with random degrees of freedom are considered. Moreover, some tables concerning considered distribution functions are demonstrated in Appendix.