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http://dx.doi.org/10.14317/jami.2014.055

ON MARCINKIEWICZ'S TYPE LAW FOR FUZZY RANDOM SETS  

Kwon, Joong-Sung (Department of Mathematics, Sun Moon University)
Shim, Hong-Tae (Department of Mathematics, Sun Moon University)
Publication Information
Journal of applied mathematics & informatics / v.32, no.1_2, 2014 , pp. 55-60 More about this Journal
Abstract
In this paper, we will obtain Marcinkiewicz's type limit laws for fuzzy random sets as follows : Let {$X_n{\mid}n{\geq}1$} be a sequence of independent identically distributed fuzzy random sets and $E{\parallel}X_i{\parallel}^r_{{\rho_p}}$ < ${\infty}$ with $1{\leq}r{\leq}2$. Then the following are equivalent: $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ a.s. in the metric ${\rho}_p$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in probability in the metric ${\rho}_p$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in $L_1$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in $L_r$ where $S_n={\Sigma}^n_{i=1}\;X_i$.
Keywords
Random sets; Fuzzy random sets; Law of large numbers; Convergence in probability; Almost everywhere convergence; $L_1$-convergence; Embedding;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
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