• Title, Summary, Keyword: strong law

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NOTE ON STRONG LAW OF LARGE NUMBER UNDER SUB-LINEAR EXPECTATION

  • Hwang, Kyo-Shin
    • East Asian mathematical journal
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    • v.36 no.1
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    • pp.25-34
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    • 2020
  • The classical limit theorems like strong law of large numbers, central limit theorems and law of iterated logarithms are fundamental theories in probability and statistics. These limit theorems are proved under additivity of probabilities and expectations. In this paper, we investigate strong law of large numbers under sub-linear expectation which generalize the classical ones. We give strong law of large numbers under sub-linear expectation with respect to the partial sums and some conditions similar to Petrov's. It is an extension of the classical Chung type strong law of large numbers of Jardas et al.'s result. As an application, we obtain Chung's strong law of large number and Marcinkiewicz's strong law of large number for independent and identically distributed random variables under the sub-linear expectation. Here the sub-linear expectation and its related capacity are not additive.

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

  • SHEN, AITING
    • Journal of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.45-55
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    • 2016
  • Let {$X_n,n{\geq}1$} be a sequence of negatively superadditive dependent random variables. In the paper, we study the strong law of large numbers for general weighted sums ${\frac{1}{g(n)}}{\sum_{i=1}^{n}}{\frac{X_i}{h(i)}}$ of negatively superadditive dependent random variables with non-identical distribution. Some sufficient conditions for the strong law of large numbers are provided. As applications, the Kolmogorov strong law of large numbers and Marcinkiewicz-Zygmund strong law of large numbers for negatively superadditive dependent random variables are obtained. Our results generalize the corresponding ones for independent random variables and negatively associated random variables.

STRONG LAW OF LARGE NUMBERS FOR ASYMPTOTICALLY NEGATIVE DEPENDENT RANDOM VARIABLES WITH APPLICATIONS

  • Kim, Hyun-Chull
    • Journal of applied mathematics & informatics
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    • v.29 no.1_2
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    • pp.201-210
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    • 2011
  • In this paper, we obtain the H$\`{a}$jeck-R$\`{e}$nyi type inequality and the strong law of large numbers for asymptotically linear negative quadrant dependent random variables by using this inequality. We also give the strong law of large numbers for the linear process under asymptotically linear negative quadrant dependence assumption.

ON THE STRONG LAW OF LARGE NUMBERS FOR LINEARLY POSITIVE QUADRANT DEPENDENT RANDOM VARIABLES

  • Kim, Tae-Sung;Seo, Hye-Young
    • Communications of the Korean Mathematical Society
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    • v.13 no.1
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    • pp.151-158
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    • 1998
  • In this note we derive inequalities of linearly positive quadrant dependent random variables and obtain a strong law of large numbers for linealy positive quardant dependent random variables. Our results imply an extension of Birkel's strong law of large numbers for associated random variables to the linear positive quadrant dependence case.

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A UNIFORM STRONG LAW OF LARGE NUMBERS FOR PARTIAL SUM PROCESSES OF FUZZY RANDOM SETS

  • Kwon, Joong-Sung;Shim, Hong-Tae
    • Journal of applied mathematics & informatics
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    • v.30 no.3_4
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    • pp.647-653
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    • 2012
  • In this paper, we consider fuzzy random sets as (measurable) mappings from a probability space into the set of fuzzy sets and prove a uniform strong law of large numbers for sequences of independent and identically distributed fuzzy random sets. Our results generalize those of Bass and Pyke(1984)and Jang and Kwon(1998).