• The Journal of Natural Sciences
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• v.13 no.1
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• pp.1-6
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• 2003

Strong laws of large numbers are established for the weighted sums of i.i.d. random variables which have higher order moment condition. One of the results of Sung(2001) is extended.

• Bulletin of the Korean Mathematical Society
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• v.33 no.3
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• pp.419-425
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• 1996

Let ${X, X_n, n \geq 1}$ be a sequence of independent and identically distributed(i.i.d) random variables with EX = 0 and $E$\mid$X$\mid$^p < \infty$ for some $p \geq 1$. Let ${a_{ni}, 1 \leq i \leq n, n \geq 1}$ be a triangular arrary of constants. The almost sure(a.s) convergence of weighted sums $\sum_{i=1}^{n} a_{ni}X_i$ can be founded in Choi and Sung[1], Chow[2], Chow and Lai[3], Li et al. [4], Stout[6], Sung[8], Teicher[9], and Thrum[10].

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.897-907
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• 2017

In this paper, we consider the strong stability of a type of Jamison weighted sums, which not only extend the corresponding result of Jamison etc. [13] from i.i.d. case to END random variables, but also obtain the necessary and sufficient results. As an important consequence, we present the result of SLLN as that of i.i.d. case.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.57-67
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• 1997

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.771-778
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• 2006

Strong laws are established for linear statistics that are weighted sums of a random sample. We show extensions of the Marcinkiewicz-Zygmund strong laws under certain moment conditions on both the weights and the distribution. The result obtained extends and sharpens the result of Sung ([12]).

• Bulletin of the Korean Mathematical Society
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• v.46 no.4
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• pp.617-626
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• 2009

Let {$Y_i$,-$\infty$ < i < $\infty$} be a doubly infinite sequence of i.i.d. random variables with E|$Y_1$| < $\infty$, {$a_{ni}$,-$\infty$ < i < $\infty$ n $\geq$ 1} an array of real numbers. Under some conditions on {$a_{ni}$}, we obtain necessary and sufficient conditions for $\sum\;_{n=1}^{\infty}\frac{1}{n}P(|\sum\;_{i=-\infty}^{\infty}a_{ni}(Y_i-EY_i)|$>$n{\epsilon})$<{\infty}$. We examine whether the result of Spitzer [11] holds for the moving average process, and give a partial solution.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.607-615
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• 2002

Let (X, $X_{n}$, n$\geq$1) be a sequence of i.i.d. random variables and { $a_{ni}$ , 1$\leq$i$\leq$n, n$\geq$1} be an array of constants. Let ø($\chi$) be a positive increasing function on (0, $\infty$) satisfying ø($\chi$) ↑ $\infty$ and ø(C$\chi$) = O(ø($\chi$)) for any C > 0. When EX = 0 and E[ø(｜X｜)]〈$\infty$, some conditions on ø and { $a_{ni}$ } are given under which (equation omitted).).

• Journal of the Korean Statistical Society
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• v.34 no.4
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• pp.263-272
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• 2005

Let {$X_n,\;n{\ge}1$} be a sequence of negatively associated random variables which are dominated randomly by another random variable. We discuss the limit properties of weighted sums ${\Sigma}^n_{i=1}a_{ni}X_i$ under some appropriate conditions, where {$a_{ni},\;1{\le}\;i\;{\le}\;n,\;n\;{\ge}\;1$} is an array of constants. As corollary, the results of Bai and Cheng (2000) and Sung (2001) are extended from the i.i.d. case to not necessarily identically distributed negatively associated setting. The corresponding results of Chow and Lai (1973) also are extended.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.241-252
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• 2012

Let $\{X_{ni},\;1{\leq}i{\leq}n,\;n{\geq}1\}$ be a sequence of LNQD which are dominated randomly by another random variable X. We obtain the complete convergence and almost sure convergence of weighted sums ${\sum}^n_{i=1}a_{ni}X_{ni}$ for LNQD by using a new exponential inequality, where $\{a_{ni},\;1{\leq}i{\leq}n,\;n{\geq}1\}$ is an array of constants. As corollary, the results of some authors are extended from i.i.d. case to not necessarily identically LNQD case.

• Journal of the Korean Statistical Society
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• v.34 no.1
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• pp.21-33
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• 2005

Let ｛X/sun ni/ ｜ 1 ≤ i ≤ n, n ≥ 1 ｝ be an array of rowwise negatively associated random variables. We in this paper discuss the conditions of n/sup -1/p/ (equation omitted) →0 completely as n → ∞ for some 1 ≤ p < 2 under not necessarily identically distributed setting. As application, it is obtained that n/sup -1/p/ (equation omitted) →0 completely as n → ∞ if and only if E｜X/sub 11/｜/sup 2p/ < ∞ and EX/sub ni=0 under identically distributed case such that the corresponding results on i. i. d. case are extended and the strong convergence for weighted sums of rowwise negatively associated arrays is also considered.