• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.341-349
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• 2003

Let {$x_{nk}\;$\mid$1\;\leq\;k\;\leq\;n,\;n\;\geq\;1$} be an array of random varianbles and $\{a_n$\mid$n\;\geq\;1\}\;and\;\{b_n$\mid$n\;\geq\;1} be a sequence of constants with $a_n\;>\;0,\;b_n\;>\;0,\;n\;\geq\;1. In this paper, for array of row negatively associated(NA) random variables, we establish a general weak law of large numbers (WLLA) of the form (${\sum_{\kappa=1}}^n\;a_{\kappa}X_{n\kappa}\;-\;\nu_{n\kappa})\;/b_n$ converges in probability to zero, as $n\;\rightarrow\;\infty$, where {$\nu_{n\kappa}$\mid$1\;\leq\;\kappa\;\leq\;n,\;n\;\geq\;1$} is a suitable array of constants.

• Journal of Positioning, Navigation, and Timing
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• v.7 no.2
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• pp.53-59
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• 2018

This paper proposes a pseudo-random beamforming technique for time-synchronized mobile base stations (BSs) for multi-cell downlink networks which have mobility. The base stations equipped with multi-antennas and mobile stations (MSs) are time-synchronized based on global positioning system (GPS) signals and generate a number of transmit beamforming matrix candidates according to the predetermined pseudo-random pattern. In addition, MSs generate receive beamforming vectors that correspond to the beam index number based on the minimum mean square error (MMSE) using transmit beamforming vectors that make up a number of transmit beamforming matrices and wireless channel matrices from BSs estimated via the reference signals (RS). Afterward, values of received signal-to-interference-plus-noise ratio (SINR) with regard to all transmit beamforming vectors are calculated, and the resulting values are then feedbacked to the BS of the same cells along with the beam index number. Each of the BSs calculates each of the sum-rates of the transmit beamforming matrix candidates based on the feedback information and then transmits the calculated results to the BS coordinator. After this, optimum transmit beamforming matrices, which can maximize a sum-rate of the entire cells, are selected at the BS coordinator and informed to the BSs. Finally, data signals are transmitted using them. The simulation results verified that a sum-rate of the entire cells was improved as the number of transmit beamforming matrix candidates increased. It was also found that if the received SINR values and beam index numbers are feedbacked opportunistically from each of the MSs to the BSs, not only nearly the same performance in sum-rate with that of applying existing feedback techniques could be achieved but also an amount of feedback was significantly reduced.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.779-786
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• 2006

Let $\mathbb{X}_t$ be an m-dimensional linear process defined by $\mathbb{X}_t=\sum{_{j=0}^\infty}\;A_j\;\mathbb{Z}_{t-j}$, t = 1, 2, $\ldots$, where $\mathbb{Z}_t$ is a sequence of m-dimensional random vectors with mean 0 : $m\times1$ and positive definite covariance matrix $\Gamma:m{\times}m$ and $\{A_j\}$ is a sequence of coefficient matrices. In this paper we give sufficient conditions so that $\sum{_{t=1}^{[ns]}\mathbb{X}_t$ (properly normalized) converges weakly to Wiener measure if the corresponding result for $\sum{_{t=1}^{[ns]}\mathbb{Z}_t$ is true.

• The Journal of Natural Sciences
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• v.9 no.1
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• pp.57-60
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• 1997

Let {$X_{ni}$,1$\leq$i$\leq$,n$\geq$1} be an array of rowwise independent B-valued random variables which is uniformly bounded by a random various X satisfying $E|X|^{2p}<\infty$ for some p$\geq$1. Let {$a_{ni}$,1$\leq$i$\leq$,n$\geq$1} be an array of constants. Under some auxiliary conditions on {$a_{ni}$}, it is shown that $sum_{i=1}^n a_{ni}X_{ni}\rightarrow0$ in probability if and only if $sum_{i=1}^n a_{ni}X_{ni}$ converges completely ot 0.

• Proceedings of the Korean Statistical Society Conference
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• 2002.05a
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• pp.49-54
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• 2002

For arbitrary random variables {$X_{n},n{\geq}1$}, the order of growth of the series. $S_{n}\;=\;{\sum}_{j=1}^n\;X_{j}$ is studied in this paper. More specifically, when the series S_{n}$ diverges almost surely, the strong law of large numbers $S_{n}/g_{n}^{-1}$($A_{n}{\psi}(A_{n}))\;{\rightarrow}\;0$ a.s. is constructed by extending the results of Petrov (1973). On the other hand, if the series $S_{n}$ converges almost surely to a random variable S, then the tail series $T_{n}\;=\;S\;-\;S_{n-1}\;=\;{\sum}_{j=n}^{\infty}\;X_{j}$ is a well-defined sequence of random variables and converges to 0 almost surely. For the almost surely convergent series $S_{n}$, a tail series strong law of large numbers $T_{n}/g_{n}^{-1}(B_{n}{\psi}^{\ast}(B_{n}^{-1}))\;{\rightarrow}\;0$ a.s., which generalizes the result of Klesov (1984), is also established by investigating the duality between the limiting behavior of partial sums and that of tail series. In particular, an example is provided showing that the current work can prevail despite the fact that previous tail series strong law of large numbers does not work.

• ETRI Journal
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• v.34 no.6
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• pp.879-884
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• 2012

When the number of users is finite, the performance improvement of the orthogonal random beamforming (ORBF) scheme is limited in high signal-to-noise ratio regions. In this paper, to improve the performance of the ORBF scheme, the user set and transmit power allocation are jointly determined to maximize sum rate under the total transmit power constraint. First, the transmit power allocation problem is expressed as a function of a given user set. Based on this expression, the optimal user set with the maximum sum rate is determined. The suboptimal procedure is also presented to reduce the computational complexity, which separates the user set selection procedure and transmit power allocation procedure.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.993-1005
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• 2008

Let {$X,\;X_n;n{\geq}1$} be a sequence of i.i.d. random variables. Set $S_n=X_1+X_2+{\cdots}+X_n,\;M_n=\max_{k{\leq}n}|S_k|,\;n{\geq}1$. Then we obtain that for any -1$\lim\limits_{{\varepsilon}{\searrow}0}\;{\varepsilon}^{2b+2}\sum\limits_{n=1}^\infty\;{\frac {(log\;n)^b}{n^{3/2}}\;E\{M_n-{\varepsilon}{\sigma}\sqrt{n\;log\;n\}+=\frac{2\sigma}{(b+1)(2b+3)}\;E|N|^{2b+3}\sum\limits_{k=0}^\infty\;{\frac{(-1)^k}{(2k+1)^{2b+3}$ if and only if EX=0 and $EX^2={\sigma}^2<{\infty}$.

• 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.

• Communications of the Korean Mathematical Society
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• v.23 no.1
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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.

• The Journal of Natural Sciences
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• v.11 no.1
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• pp.15-17
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• 1999

Let {f(n)} be an increasing sequence such that f(n)>0 for each n and f(n)$\rightarrow$$\infty$. Let {X$_n$,n$\geq1$} be a sequence of pairwise independent random variables. In this paper we give sufficient conditions on {X$_n$,n$\geq1$} such that $sum_{i=1}^n$(X$_i$-EX$_i$)/f(n) converges to zero almost surely.