• 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.

• The Journal of the Korea Contents Association
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• v.6 no.4
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• pp.63-68
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• 2006

For the almost co티am convergent series $S_n$ of independent random variables, by investigating the limiting behavior of the tail series, $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}X_i$, the rate of convergence of the series $S_n$ to a random variable S is studied in this paper. More specifically, the equivalence between the tail series weak law of large numbers and a limit law is established for a quasi-monotone decreasing sequence, thereby extending a result of Previous work to the wider class of the norming constants.

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.531-538
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• 2005

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

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.377-384
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• 1998

Let ${X_n,n \geq 1}$ be a sequence of stochatically dominated pairwise independent random variables. Let ${a_n, n \geq 1}$ and ${b_n, n \geq 1}$ be seqence of constants such that $a_n \neq 0$ and $0 < b_n \uparrow \infty$. A strong law large numbers of the form $\sum^{n}_{j=1}{a_j X_i//b_n \to 0$ almost surely is obtained.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.321-328
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• 1995

Ibragimov(1975) showed the central limit theorem and the invariance principle for $\rho$-mixing random variables satisfying $\sigma^2(n) = nh(n) \longrightarrow \infty$ and $E$\mid$\zeta_0$\mid$^{2+\delta} < \infty$ for some $\delta > 0$ where $\sigma^2(n)$ denotes the variance of the partial sum $S_n$ and h(n) is a slowly varying function.

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.255-263
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• 2000

Let ${X_{nk}}, u_n\; \leq \;k \leq \;u_n,\; n\; \geq\; 1}$ be an array of rowwise independent, but not necessarily identically distributed, random variables with $EX_{nk}$=0 for all k and n. In this paper, we povide a domination condition under which ${\sum^{u_n}}_=u_n\; S_{nk}/n^{1/p},\; 1\; \leq\; p\;<2$ converges completely to zero.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.221-230
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• 2007

The exponential and the gamma distributions have been the traditional models for drought duration and drought intensity data, respectively. However, it is often assumed that the drought duration and drought intensity are independent, which is not true in practice. In this paper, an application of the bivariate gamma exponential distribution is provided to drought data from Nebraska. The exact distributions of R=X+Y, P=XY and W=X/(X+Y) and the corresponding moment properties are derived when X and Y follow this bivariate distribution.

• Journal of the Korean Statistical Society
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• v.33 no.1
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• pp.99-106
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• 2004

Let {X, $X_{n}, n\;{\geq}\;1$} be a sequence of identically distributed, negatively associated (NA) random variables and assume that $│X│^{r}$, r > 0, has a finite moment generating function. A strong law of large numbers is established for weighted sums of these variables.

• Journal of applied mathematics & informatics
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• v.39 no.1_2
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• pp.145-153
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• 2021

When {Xni|1 ≤ i ≤ n, n ≥ 1} be an array of rowwise negatively superadditive dependent(NSD) for semi-Gaussian random variables and {ani|1 ≤ i ≤ n, n ≥ 1} is an array of constants, we study the almost sure convergence of weighted sums ∑ni=1 aniXni under some appropriate conditions and we obtain some corollaries.

• 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.