• 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.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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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 ${\epsilon}{\searrow}0$.

• 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.

• Journal of Communications and Networks
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• v.11 no.5
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• pp.480-488
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• 2009

Cooperative transmission is an effective solution to improve the performance of wireless communications over fading channels without the need for physical co-located antenna arrays. In this paper, selection combining is used at the destination instead of maximal ratio combing to optimize the structure of destination and to reduce power consumption in selective decode-and-forward relaying networks. For an arbitrary number of relays, an exact and closed-form expression of the bit error rate (BER) is derived for M-PAM, M-QAM, and M-PSK, respectively, in both independent identically distributed and independent but not identically distributed Rayleigh fading channels. A variety of simulations are performed and show that they match exactly with analytic ones. In addition, our results show that the optimum number of relays depend not only on channel conditions (operating SNRs) but also on modulation schemes which to be used.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.505-513
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• 2009

Let {$X_k;k{\in}N^d$} be centered and identically distributed random field which is asymptotically negative dependent in a certain case. In this note we prove that for $p{\alpha}$ > 1 and ${\alpha}$ > ${\frac{1}{2}}$ $E{\mid}X_1{\mid}^p(log^+{\mid}X_1{\mid}^{d-1})$ < ${\infty}$ if and only if ${\sum}_n{\mid}n{\mid}^{p{\alpha}-2}P$($max_{1{\leq}k{\leq}n{\mid}S_k{\mid}}$ > ${\epsilon}{\mid}n{\mid}$) < ${\infty}$ for all ${\epsilon}$ > 0, where log$^+$x = max{1,log x}.

• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.243-254
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• 2006

The relative frequency of order statistics is investigated for independent and identically distributed (i.i.d.) random variables. Specifically, it is shown that the probability $Pr\{X_{[s]}=x\}$ is no less than the probability $Pr\{X_{[r]}=x\}$ at any point $x{\geqq}x_0$ when r$X_{[r]}$ denotes the r-th order statistic of an i.i.d. discrete random vector and $x_0$ depends on the population probability distribution. A similar result for i.i.d. continuous random vectors is also presented.

• 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.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.119-128
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• 2010

Let ${X_n,\;n\geq1}$ be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space ($\Omega$,A,P), and let ${N_n,\;n\geq1}$ be a sequence of positive integer-valued r.vs., defined on the same probability space ($\Omega$,A,P). Furthermore, we assume that the r.vs. $N_n$, $n\geq1$ are independent of all r.vs. $X_n$, $n\geq1$. In present paper we are interested in asymptotic behaviors of the random sum $S_{N_n}=X_1+X_2+\cdots+X_{N_n}$, $S_0=0$, where the r.vs. $N_n$, $n\geq1$ obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums $S_{N_n}$ have been investigated in the theory probability and stochastic processes for quite some time (see [1], [4], [2], [3], [5]). Recently, the random sum approach is used in some applied problems of stochastic processes, stochastic modeling, random walk, queue theory, theory of network or theory of estimation (see [10], [12]). The main aim of this paper is to establish some results related to the asymptotic behaviors of the random sum $S_{N_n}$, in cases when the $N_n$, $n\geq1$ are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.

• Journal of Communications and Networks
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• v.1 no.2
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• pp.111-117
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• 1999

Using a method recently reported in the literature for analyzing the bit error probability (BEP) performance of noncoherent Mary orthogonal signals with square-law combining in the presence of independent and identically distributed Nakagami-m faded paths, we are able to reformulate this method so as to apply to a generalized fading channel in which the fading in each path need not be identically distributed nor even distributed ac-cording to the same family of distribution. The method leads to exact expressions for the BEP in the form of a finite-range integral whose integrand involves the moment generating function of the combined signal-to-noise ratio and which can therefore be readily evaluated numerically. The mathematical formalism is illustrated by applying the method to some selected numerical examples of interest showing the impact of the multipath intensity profile (MIP) as well as the fading correlation profile (FCP) on the BEP performance of M-ary orthogonal signal over Nakagami-m fading channels. Thses numerical results show that both MIP and FCP induce a non-negligible degradition in the BEP and have therefore to be taken into account for the accurate prediction of the performance of such systems.

• Journal of the Korean Statistical Society
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• v.13 no.2
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• pp.81-86
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• 1984

Under the i.i.d. hypothesis, authors (1982, 1984) proved some local limit theorems both for the continuous case and for the lattice case. In this paper, results are extended to the case where the random vectors are not identically distributed.