scholarly journals General theorems on rates of convergence in distribution of random variables II. Applications to the stable limit laws and weak law of large numbers

1978 ◽  
Vol 8 (2) ◽  
pp. 202-221 ◽  
Author(s):  
P.L. Butzer ◽  
L. Hahn
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Stable Limit ◽  
Convergence In Distribution ◽  
Limit Laws ◽  
Large Numbers

Obtaining Prescribed Rates of Convergence for the Ergodic Theorem

1983 ◽  
Vol 35 (6) ◽  
pp. 1129-1146 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Ergodic Theorem ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Stationary Sequence ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Image Position

Let {Yn, n ∊ Z} be an ergodic strictly stationary sequence of random variables with mean zero, where Z denotes the set of integers. For n ∊ N = {1, 2, …}, let Sn = Y1 + Y2 + … + Yn. The ergodic theorem, alias the strong law of large numbers, says that n–lSn → 0 as n → ∞ a.s. If the Yn's are independent and have variance one, the law of the iterated logarithm tells us that this convergence takes place at the rate in the sense that1It is our purpose here to investigate what other rates of convergence are possible for the ergodic theorem, that is to say, what sequences {bn, n ≧ 1} have the property that2for some ergodic stationary sequence {Yn, n ∊ Z}.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

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