scholarly journals On the laws of the iterated logarithm under sub-linear expectations

2021 ◽  
Vol 6 (4) ◽  
pp. 409
Author(s):  
Li-Xin Zhang
Keyword(s):  
Necessary Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Exponential Inequalities ◽  
Iterated Logarithm ◽  
Sufficient And Necessary Conditions ◽  
The Law ◽  
Independent And Identically Distributed

<p style='text-indent:20px;'>In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.</p>

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

scholarly journals On Feller's criterion for the law of the iterated logarithm

1994 ◽  
Vol 17 (2) ◽  
pp. 323-340 ◽  
Author(s):  
Deli Li ◽  
M. Bhaskara Rao ◽  
Xiangchen Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Uniform Estimate ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Combining Feller's criterion with a non-uniform estimate result in the context of the Central Limit Theorem for partial sums of independent random variables, we obtain several results on the Law of the Iterated Logarithm. Two of these results refine corresponding results of Wittmann (1985) and Egorov (1971). In addition, these results are compared with the corresponding results of Teicher (1974), Tomkins (1983) and Tomkins (1990)

On the converse to the iterated logarithm law

1968 ◽  
Vol 5 (01) ◽  
pp. 210-215 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Variables ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
The Law ◽  
Image Position ◽  
Iterated Logarithm Law ◽  
Independent And Identically Distributed

Let Xi, i = 1, 2, 3,… be a sequence of independent and identically distributed random variables with law ℓ(X) and write. if EX = 0 and EX2 = σ2 &lt; ∞, the law of the iterated logarithm (Hartman and Wintner [1]) tells us that

scholarly journals On the rate of convergence in the strong law of large numbers for arrays

1992 ◽  
Vol 45 (3) ◽  
pp. 479-482 ◽  
Author(s):  
Tien-Chung Hu ◽  
N.C. Weber
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Second Moment ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the law of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finite fourth moments.

scholarly journals The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1313
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Independent Random Variable ◽  
Random Variables ◽  
Random Variable ◽  
Second Order ◽  
Law Of Iterated Logarithm ◽  
Independent Random Variables ◽  
Linear Processes ◽  
Iterated Logarithm ◽  
The Law

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

scholarly journals On strong convergence of arrays

1994 ◽  
Vol 50 (2) ◽  
pp. 219-223 ◽  
Author(s):  
Yong-Cheng Qi
Keyword(s):  
Strong Convergence ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Strong Law ◽  
Iterated Logarithm ◽  
The Law ◽  
Independent And Identically Distributed

In this paper we study almost sure convergence for arrays of independent and identically distributed random variables. We obtain a condition under which Marcinkiewicz's strong law holds and get a rate analogous to the law of the iterated logarithm under a condition weaker than Hu and Weber's.

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