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POWER INEQUALITY ON THE SIMPLEX
백인수 충청수학회 2012 충청수학회지 Vol.25 No.2
The power inequality QN k=1 xxk k QN k=1 pxk k holds for the points (x1; :::; xN),(p1; :::; pN) of the simplex. We show this using the analytic method combining Frostman's density theorem with the strong law of large numbers.
IDENTICALLY DISTRIBUTED UNCORRELATED RANDOM VARIABLES NOT FULFILLING THE WLLN
Landers, Dieter,Rogge, Lothar Korean Mathematical Society 2001 대한수학회보 Vol.38 No.3
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.
SOME NOTES ON STRONG LAW OF LARGE NUMBERS FOR BANACH SPACE VALUED FUZZY RANDOM VARIABLES
Kim, Joo-Mok,Kim, Yun Kyong The Kangwon-Kyungki Mathematical Society 2013 한국수학논문집 Vol.21 No.4
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 EXPONENTIAL INEQUALITY FOR NEGATIVE DEPENDENT SEQUENCE
Kim, Tae-Sung,Kim, Hyun-Chull Korean Mathematical Society 2007 대한수학회논문집 Vol.22 No.2
We show an exponential inequality for negatively associated and strictly stationary random variables replacing an uniform boundedness assumption by the existence of Laplace transforms. To obtain this result we use a truncation technique together with a block decomposition of the sums. We also identify a convergence rate for the strong law of large number.
Note on Strong Law of Large Number under Sub-linear Expectation
황교신 영남수학회 2020 East Asian mathematical journal Vol.36 No.1
The classical limit theorems like strong law of large numbers, central limit theorems and law of iterated logarithms are fundamental the- ories 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 sim- ilar 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 ob- tain Chung’s strong law of large number and Marcinkiewicz’s strong law of large number for independent and identically distributed random vari- ables under the sub-linear expectation. Here the sub-linear expectation and its related capacity are not additive.
SHEN, AITING Korean Mathematical Society 2016 대한수학회지 Vol.53 No.1
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.
Aiting Shen 대한수학회 2016 대한수학회지 Vol.53 No.1
Let $\{X_n, n\geq1\}$ 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.
NOTE ON STRONG LAW OF LARGE NUMBER UNDER SUB-LINEAR EXPECTATION
Hwang, Kyo-Shin The Youngnam Mathematical Society 2020 East Asian mathematical journal Vol.36 No.1
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.
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. Korean Mathematical Society 1998 대한수학회논문집 Vol.13 No.4
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.
STRONG LAW OF LARGE NUMBERS FOR ASYMPTOTICALLY NEGATIVE DEPENDENT RANDOM VARIABLES WITH APPLICATIONS
Kim, Hyun-Chull The Korean Society for Computational and Applied M 2011 Journal of applied mathematics & informatics Vol.29 No.1
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.