• Title/Summary/Keyword: uniformly integrable

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THE UNIFORM CLT FOR MARTINGALE DIFFERENCE ARRAYS UNDER THE UNIFORMLY INTEGRABLE ENTROPY

  • Bae, Jong-Sig;Jun, Doo-Bae;Levental, Shlomo
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.39-51
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    • 2010
  • In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler [19] did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler [19] and other results of independent problems. The results also generalizes those of Bae and Choi [3] to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.

THE CONVERGENCE THEOREMS FOR THE McSHANE-STIELTJES INTEGRAL

  • Yoon, Ju-Han;Kim, Byung-Moo
    • The Pure and Applied Mathematics
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    • v.7 no.2
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    • pp.137-143
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    • 2000
  • In this paper, we define the uniformly sequence for the vector valued McShand-Stieltjes integrable functions and prove the dominated convergence theorem for the McShand-Stieltjes integrable functions.

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STRONG CONVERGENCE FOR WEIGHTED SUMS OF FUZZY RANDOM VARIABLES

  • Kim, Yun-Kyong
    • Proceedings of the Korean Statistical Society Conference
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    • 2003.10a
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    • pp.183-188
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    • 2003
  • In this paper, we establish some results on strong convergence for weighted sums of uniformly integrable fuzzy random variables taking values in the space of upper-semicontinuous fuzzy sets in R$^{p}$.

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Random Elements in $L^1(R)$ and Kernel Density Estimators

  • Lee, Sung-Ho;Lee, Robert -Taylor
    • Journal of the Korean Statistical Society
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    • v.22 no.1
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    • pp.83-91
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    • 1993
  • Random elements in $L^1(R)$ and some properties of $L^1(R)$ space are investigated with application to kernel density estimators. A weak law of large numbers for compact uniformly integrable random elements is introduced for further application.

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A Weak Convergence Theorem for Mixingale Arrays

  • Hong, Dug-Hun;Kim, Hye-Kyung;Kim, Ju-Young
    • Journal of the Korean Statistical Society
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    • v.24 no.2
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    • pp.273-280
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    • 1995
  • This paper gives a generalization of an $L_1$-convergence theorem for dependent processes due to Andrews (1988) and also a probability convergence theorem.

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STRONG LAWS OF LARGE NUMBERS FOR RANDOM UPPER-SEMICONTINUOUS FUZZY SETS

  • Kim, Yun-Kyong
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.511-526
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    • 2002
  • In this paper, we concern with SLLN for sums Of in-dependent random upper-semicontinuous fuzzy sets. We first give a generalization of SLLN for sums of independent and level-wise identically distributed random fuzzy sets, and establish a SLLN for sums of random fuzzy sets which is independent and compactly uniformly integrable in the strong sense. As a result, a SLLN for sums of independent and strongly tight random fuzzy sets is obtained.