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ON MARCINKIEWICZ'S TYPE LAW FOR FUZZY RANDOM SETS
ON MARCINKIEWICZ'S TYPE LAW FOR FUZZY RANDOM SETS
Journal of applied mathematics & informatics. 2014. Jan, 32(1_2): 55-60
  • Received : July 03, 2013
  • Published : January 30, 2014
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Kwon Joong-Sung
Shim Hong-Tae

Abstract
In this paper, we will obtain Marcinkiewicz's type limit laws for fuzzy random sets as follows : Let {<TEX>$X_n{\mid}n{\geq}1$</TEX>} be a sequence of independent identically distributed fuzzy random sets and <TEX>$E{\parallel}X_i{\parallel}^r_{{\rho_p}}$</TEX> < <TEX>${\infty}$</TEX> with <TEX>$1{\leq}r{\leq}2$</TEX>. Then the following are equivalent: <TEX>$S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$</TEX> a.s. in the metric <TEX>${\rho}_p$</TEX> if and only if <TEX>$S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$</TEX> in probability in the metric <TEX>${\rho}_p$</TEX> if and only if <TEX>$S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$</TEX> in <TEX>$L_1$</TEX> if and only if <TEX>$S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$</TEX> in <TEX>$L_r$</TEX> where <TEX>$S_n
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