• 대한수학회논문집
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• 제23권1호
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• pp.133-140
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• 2008

Let {${\xi}_k,\;k\;{\in}\;{\mathbb{Z}}$} be a strictly stationary associated sequence of H-valued random variables with $E{\xi}_k\;=\;0$ and $E{\parallel}{\xi}_k{\parallel}^2\;<\;{\infty}$ and {$a_k,\;k\;{\in}\;{\mathbb{Z}}$} a sequence of linear operators such that ${\sum}_{j=-{\infty}}^{\infty}\;{\parallel}a_j{\parallel}_{L(H)}\;<\;{\infty}$. For a linear process $X_k\;=\;{\sum}_{j=-{\infty}}^{\infty}\;a_j{\xi}_{k-j}$ we derive that {$X_k} fulfills the functional central limit theorem.

• 대한수학회지
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• 제41권5호
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• pp.883-894
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• 2004

We discuss in this paper the strong convergence for weighted sums of negative associated (in abbreviation: NA) arrays. Meanwhile, the central limit theorem for weighted sums of NA variables and linear process based on NA variables is also considered. As corollary, we get the results on iid of Li et al. ([10]) in NA setting.

• Communications for Statistical Applications and Methods
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• 제13권2호
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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.

• 대한수학회보
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• 제47권3호
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• pp.467-482
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• 2010

For a double array of random elements {$V_{mn};m{\geq}1,\;n{\geq}1$} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which $k_{mn}^{-\frac{1}{r}}\sum{{u_m}\atop{i=1}}\sum{{u_n}\atop{i=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in $L_r$ (0 < r < 2). The weak law results provide conditions for $k_{mn}^{-\frac{1}{r}}\sum{{T_m}\atop{i=1}}\sum{{\tau}_n\atop{j=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in probability where {$T_m;m\;{\geq}1$} and {${\tau}_n;n\;{\geq}1$} are sequences of positive integer-valued random variables, {$k_{mn};m{{\geq}}1,\;n{\geq}1$} is an array of positive integers. The sharpness of the results is illustrated by examples.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.1025-1034
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• 2010

Let {$X_n$, $n\;{\in}\;\mathbb{Z}_+^d$} be a field of identically distributed and negatively associated random variables with mean zero and set $S_n\;=\;{\sum}_{k{\leq}n}\;X_k$, $n\;{\in}\;\mathbb{Z}_+^d$, $d\;{\geq}\;2$. We investigate precise asymptotics for ${\sum}_n|n|^{r/p-2}P(|S_n|\;{\geq}\;{\epsilon}|n|^{1/p}$ and ${\sum}_n\;\frac{(\log\;|n|)^{\delta}}{|n|}P(|S_n|\;{\geq}\;{\epsilon}\;\sqrt{|n|\log|n|)}$, ($0\;{\leq}\;{\delta}\;{\leq}\;1$) as ${\epsilon}{\searrow}0$.

• Communications for Statistical Applications and Methods
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• 제16권3호
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• pp.549-556
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• 2009

Let $\{(X_{ni}|1{\leq}i{\leq}n,\;n{\geq}1)\}$ be an array of rowwise negatively associated random variables. In this paper we discuss $n^{{\alpha}p-2}h(n)max_{1{\leq}k{\leq}n}|{\sum}_{i=1}^kX_{ni}|/n^{\alpha}{\rightarrow}0$ completely as $n{\rightarrow}{\infty}$ under not necessarily identically distributed with suitable conditions for ${\alpha}$>1/2, 0${\alpha}p{\geq}1$ and a slowly varying function h(x)>0 as $x{\rightarrow}{\infty}$. In addition, we obtain the complete convergence of moving average process based on negative association random variables which extends the result of Zhang (1996).

• Journal of the Korean Statistical Society
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• 제21권2호
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• pp.167-177
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• 1992

In this paper, we investigate the limiting distribution of $M_n = max (X_1, X-2, \cdots, X_n)$ in the infinite moving average process ${X_t = \sum c_i Z_{t-i}}$ generated from i.i.d. negative binomial variables $Z_i$'s. While no limit result is possible, nonetheless asymptotic bounds are derived. We also present the tail behavior of $X_t$, i.e., weighted sum of i.i.d. random variables. This continues a study made by Rootzen (1986) for discrete innovation sequences.

• Journal of applied mathematics & informatics
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• 제27권3_4호
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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.

• International Journal of Aeronautical and Space Sciences
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• 제5권2호
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• pp.1-8
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• 2004

Real-Time Kinematic GPS positioning is widely used for many applications.Resolving ambiguities is the key to precise positioning. Integer ambiguity resolution isthe process of resolving the unknown cycle ambiguities of double difference carrierphase data as integers. Two important issues of resolving are efficiency andreliability. In the conventional search techniques, we generally used chi-squarerandom variables for decision variables. Mathematically, a chi-square random variableis the sum of mutually independent, squared zero-mean unit-variance normal(Gaussian) random variables. With this base knowledge, we can separate decisionvariables to several normal random variables. We showed it with related equationsand conceptual diagrams. With this separation, we can improve the computationalefficiency of the process without losing the needed performance. If we averageseparated normal random variables sequentially, averaged values are also normalrandom variables. So we can use them as decision variables, which prevent from asudden increase of some decision variable. With the method using averaged decisionvalues, we can get the solution more quicklv and more reliably.To verify the performance of our proposed algorithm, we conducted simulations.We used some visual diagrams that are useful for intuitional approach. We analyzedthe performance of the proposed algorithm and compared it to the conventionalmethods.

• 한국콘텐츠학회논문지
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• 제6권4호
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• pp.63-68
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• 2006

본 연구에서는, 서로 독립인 확률변수들의 합 $S_n$이 수렴하는 경우에, 확률변수들의 Tail 합 $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}X_i$의 극한 성질을 연구함으로써, $S_n$이 하나의 확률변수 S로 수렴하는 속도를 연구한다. 좀 더 구체적으로 말하자면, 유사-단조감소(Quasi-monotone decreasing)하는 상수(Norming constants)의 수열에 대하여, 확률변수들의 Tail 합에 대한 약대수법칙과 하나의 수렴법칙이 동등함을 정리로 기술하고 증명하여, 기존의 연구 결과를 더 넓은 부류의 상수들의 경우에 적용할 수 있도록 확장한다.