• Honam Mathematical Journal
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• v.34 no.2
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• pp.209-217
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• 2012

Convergence rates in the law of large numbers for i.i.d. random variables have been generalized by Gut[Gut, A., 1978. Marc inkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab. 6, 469-482] to random fields with all indices having the same power in the normalization. In this paper we generalize these convergence rates to the identically distributed and negatively associated random fields with different indices having different power in the normalization.

• Communications for Statistical Applications and Methods
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• v.9 no.2
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• pp.451-457
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• 2002

We derive the strong laws of large numbers for weighted averages of partial sums of random variables which are either associated or negatively associated. Our theorems extend and generalize strong law of large numbers for weighted sums of associated and negatively associated random variables of Matula(1996; Probab. Math. Statist. 16) and some results in Birkel(1989; Statist. Probab. Lett. 7) and Matula (1992; Statist. Probab. Lett. 15 ).

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.117-126
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• 2003

Let {$X_{nk}$\mid$1\;{\leq}\;k\;{\leq}\;n,\;n\;{\geq}\;1$} be an array of row negatively associated (NA) random variables which satisfy $P($\mid$X_{nk}$\mid$\;>\;x)\;{\leq}\;P($\mid$X$\mid$\;>\;x)$. For weighed sums ${{\Sigma}_{k=1}}^{Tn}\;a_kX_{nk}$ indexed by random variables {$T_n$\mid$n\;{\geq}$1$}, we establish a general weak law of large numbers (WLLN) of the form $({{\Sigma}_{k=1}}^{Tn}\;a_kX_{nk}\;-\;v_{[nk]})\;/b_{[an]}$ under some suitable conditions, where $\{a_n$\mid$n\;\geq\;1\},\; \{b_n$\mid$n\;\geq\;1\}$ are sequences of constants with $a_n\;>\;0,\;0\;<\;b_n\;\rightarrow \;\infty,\;n\;{\geq}\;1$, and {$v_{an}$\mid$n\;{\geq}\;1$} is an array of random variables, and the symbol [x] denotes the greatest integer in x.

• Communications for Statistical Applications and Methods
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• v.12 no.1
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• pp.117-123
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• 2005

A functional central limit theorem is obtained for a stationary linear process of the form $X_{t}= \sum\limits_{j=0}^\infty{a_{j}x_{t-j}}$, where {x_t} is a strictly stationary sequence of negatively associated random variables with suitable conditions and {a_j} is a sequence of real numbers with $\sum\limits_{j=0}^\infty|a_{j}|<\infty$.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.407-415
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• 2011

We establish strong laws of large numbers for weighted sums of arrays of negatively associated random variables under the condition of h-integrability and suitable conditions on the array of weights.

• Journal of the Korean Statistical Society
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• v.31 no.3
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• pp.369-378
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• 2002

Strong laws of large numbers for weighted sums of asymptotically almost negatively associated(AANA) sequence are proved by our generalized maximal inequality for AANA random variables at a crucial step.

• 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.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.457-466
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• 2004

Let{$X_{ni}$\mid$\;1\;{\leq}\;i\;{\leq}\;k_n,\;n\;{\geq}\;1$} be an array of rowwise negatively associated random variables such that $P$\mid$X_{ni}$\mid$\;>\;x)\;=\;O(1)P($\mid$X$\mid$\;>\;x)$ for all $x\;{\geq}\;0,\;and\; \{k_n\}\;and\;\{r_n\}$ be two sequences such that $r_n\;{\geq}\;b_1n^r,\;k_n\;{\leq}\;b_2n^k$ for some $b_1,\;b_2,\;r,\;k\;>\;0$. Then it is shown that $\frac{1}{r_n}\;max_1$\mid${\Sigma_{i=1}}^j\;X_{ni}$\mid$\;{\rightarrow}\;0$ completely convergence and the strong convergence for weighted sums of N A arrays is also considered.

• 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.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.151-161
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• 2011

Let {$X_n$, $n{\geq}1$} be a sequence of asymptotically almost negatively associated random variables and $S_n=\sum^n_{i=1}X_i$. In the paper, we get the precise results of H$\acute{a}$jek-R$\acute{e}$nyi type inequalities for the partial sums of asymptotically almost negatively associated sequence, which generalize and improve the results of Theorem 2.4-Theorem 2.6 in Ko et al. ([4]). In addition, the large deviation of $S_n$ for sequence of asymptotically almost negatively associated random variables is studied. At last, the Marcinkiewicz type strong law of large numbers is given.