Exact convergence rate of an Erdös-Rényi strong law for moving quantiles

1986 ◽  
Vol 23 (2) ◽  
pp. 355-369 ◽  
Author(s):  
Paul Deheuvels ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Order Statistic ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Monotonicity Condition ◽  
Strict Monotonicity ◽  
Strong Law ◽  
Large Numbers ◽  
Image Position

Consider a sequence U1, U2, · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un+1, · ··, Un+K, and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α < u < 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X1, X2, · ·· with d.f. F satisfying a strict monotonicity condition.

Exact convergence rate of an Erdös-Rényi strong law for moving quantiles

1986 ◽  
Vol 23 (02) ◽  
pp. 355-369
Author(s):  
Paul Deheuvels ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Order Statistic ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Monotonicity Condition ◽  
Strict Monotonicity ◽  
Strong Law ◽  
Large Numbers ◽  
Image Position

Consider a sequence U 1, U 2 , · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un +1, · ··, Un +K , and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α &lt; u &lt; 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X 1, X 2, · ·· with d.f. F satisfying a strict monotonicity condition.

scholarly journals Exponential inequalities under sub-linear expectations with applications to strong law of large numbers

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2951-2961
Author(s):  
Xufei Tang ◽  
Xuejun Wang ◽  
Yi Wu
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Independent Random Variables ◽  
Exponential Inequalities ◽  
Strong Law ◽  
Large Numbers ◽  
Strong Convergence Rate

In this paper, we give some exponential inequalities for extended independent random variables under sub-linear expectations. As an application, we obtain the strong convergence rate O(n-1/2 ln1/2 n) for the strong law of large numbers under sub-linear expectations, which generalizes some corresponding ones under the classical linear expectations.

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.

A Note on the Strong Law of Large Numbers

1994 ◽  
Vol 44 (1-2) ◽  
pp. 115-122 ◽  
Author(s):  
Arup Bose ◽  
Tapas K. Chandra
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers

Let { X n} be a sequence of pairwise independent (or -mixing) mean zero random variables such that [Formula: see text] is integrable on (0,∞) and [Formula: see text] then we show that [Formula: see text] almost surely as n→∞, These are very convenient and immediate generalizations of the classical SLLN for the iid case.

scholarly journals Law of large numbers for the sequence of uncorrelated fuzzy random variables

2021 ◽  
pp. 1-8
Author(s):  
Li Guan ◽  
Jinping Zhang ◽  
Jieming Zhou
Keyword(s):  
Law Of Large Numbers ◽  
Hausdorff Metric ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Strong Law ◽  
Fuzzy Variables ◽  
Large Numbers ◽  
The Law ◽  
Fuzzy Random

This work proposes the concept of uncorrelation for fuzzy random variables, which is weaker than independence. For the sequence of uncorrelated fuzzy variables, weak and strong law of large numbers are studied under the uniform Hausdorff metric d H ∞ . The results generalize the law of large numbers for independent fuzzy random variables.

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