• Title, Summary, Keyword: strong law

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A NOTE ON THE STRONG LAW OF LARGE NUMBERS FOR WEIGHTED SUMS OF NEGATIVELY DEPENDENT RANDOM VARIABLES

  • Lee, S.W.;Kim, T.S.;Kim, H.C.
    • Communications of the Korean Mathematical Society
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    • v.13 no.4
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    • pp.855-863
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    • 1998
  • Some conditions on the strong law of large numbers for weighted sums of negative quadrant dependent random variables are studied. The almost sure convergence of weighted sums of negatively associated random variables is also established, and then it is utilized to obtain strong laws of large numbers for weighted averages of negatively associated random variables.

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SOME NOTES ON STRONG LAW OF LARGE NUMBERS FOR BANACH SPACE VALUED FUZZY RANDOM VARIABLES

  • Kim, Joo-Mok;Kim, Yun Kyong
    • Korean Journal of Mathematics
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    • v.21 no.4
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    • pp.383-399
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    • 2013
  • In this paper, we establish two types of strong law of large numbers for fuzzy random variables taking values on the space of normal and upper-semicontinuous fuzzy sets with compact support in a separable Banach space. The first result is SLLN for strong-compactly uniformly integrable fuzzy random variables, and the other is the case of that the averages of its expectations converges.

On the Strong Law of Large Numbers for Arbitrary Random Variables

  • Nam, Eun-Woo
    • Proceedings of the Korean Statistical Society Conference
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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.

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STRONG LAWS FOR ARRAYS OF RANDOM VARIABLES

  • Sung, Soo-Hak
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.769-775
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    • 1998
  • In this paper, we obtain an analogue of law of the iterated logarithm for an array of independent, but not necessarily idetically distributed, random variables under some moment conditions of the array.

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ON THE HAJECK-RENYI-TYPE INEQUALITY FOR $\tilde{\rho}$-MIXING SEQUENCES

  • Choi, Jeong-Yeol;Baek, Jong-Il
    • Honam Mathematical Journal
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    • v.30 no.3
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    • pp.479-486
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    • 2008
  • Let {${\Omega}$, F, P} be a probability space and {$X_n{\mid}n{\geq}1$} be a sequence of random variables defined on it. We study the Hajeck-Renyi-type inequality for p..mixing random variable sequences and obtain the strong law of large numbers by using this inequality. We also consider the strong law of large numbers for weighted sums of ${\tilde{\rho}}$-mixing sequences.

A STRONG LAW OF LARGE NUMBERS FOR AANA RANDOM VARIABLES IN A HILBERT SPACE AND ITS APPLICATION

  • Ko, Mi-Hwa
    • Honam Mathematical Journal
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    • v.32 no.1
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    • pp.91-99
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    • 2010
  • In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process generated by asymptotically almost negatively random variables in a Hilbert space with this result.

IDENTICALLY DISTRIBUTED UNCORRELATED RANDOM VARIABLES NOT FULFILLING THE WLLN

  • Landers, Dieter;Rogge, Lothar
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.605-610
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    • 2001
  • It is shown that for each 1 < p < 2 there exist identically distributed uncorrelated random variables $X_n\; with\;E({$\mid$X_1$\mid$}^p)\;<\;{\infty}$, not fulfilling the weak law of large numbers (WLLN). If, however, the random variables are moreover non-negative, the weaker integrability condition $E(X_1\;log\;X_1)\;<\;{\infty}$ already guarantees the strong law of large numbers.

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