Application to the Uniform Laws of Large Numbers for Dependent Processes

2019 ◽  
pp. 438-447
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Random Variables ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Single Function ◽  
Strongly Mixing ◽  
Mixing Coefficients ◽  
Mixing Sequences ◽  
Dependent Processes

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling of the initial sequence by an independent one having the same marginal. In this chapter, we shall see how the coupling results mentioned in Section 5.1.3 are very useful to derive uniform laws of large numbers for mixing sequences. The uniform laws of large numbers extend the classical laws of large numbers from a single function to a collection of such functions. We shall address this question for sequences of random variables that are either absolutely regular, or ϕ‎-mixing, or strongly mixing. In all the obtained results, no condition is imposed on the rates of convergence to zero of the mixing coefficients.

scholarly journals Laws of large numbers for infinite means with mixing sequences

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 783-793
Author(s):  
Jian Han ◽  
Xiaoqin Li ◽  
Yudan Cheng
Keyword(s):  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Mixing Sequences

In this paper, we consider the laws of large numbers with infinite means based on ?-mixing sequences. An exact weak law and a strong law are obtained for ?-mixing asymmetrical Cauchy random variables. It is also presented that the weak law cannot extend to a strong law. In addition, some simulations are presented to illustrate our results of the laws of large numbers.

scholarly journals Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables

2019 ◽  
Vol 39 (1) ◽  
pp. 19-38
Author(s):  
Shuhua Chang ◽  
Deli Li ◽  
Andrew Rosalsky
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Partial Sums ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Necessary And Sufficient

Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + . . . + Xn, n ≥ ­ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max1≤k≤n |Sk| ; n ≥ ­ 1}. More specifically, necessary and sufficient conditions are given forlimn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,where log x = loge max{e, x}, x ≥­ 0.

Laws of large numbers for q-dependent random variables and nonextensive statistical mechanics

2017 ◽  
Vol 31 (15) ◽  
pp. 1750117
Author(s):  
Marco A. S. Trindade
Keyword(s):  
Statistical Mechanics ◽  
Stochastic Processes ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Nonextensive Statistical Mechanics ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Large Numbers

In this work, we prove a weak law and a strong law of large numbers through the concept of [Formula: see text]-product for dependent random variables, in the context of nonextensive statistical mechanics. Applications for the consistency of estimators are presented and connections with stochastic processes are discussed.

scholarly journals Limiting behavior of the perturbed empirical distribution functions evaluated at U-statistics for strongly mixing sequences of random variables

1997 ◽  
Vol 10 (1) ◽  
pp. 3-20 ◽  
Author(s):  
Shan Sun ◽  
Ching-Yuan Chiang
Keyword(s):  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Functions ◽  
Limiting Behavior ◽  
U Statistics ◽  
Strongly Mixing ◽  
Almost Sure Representation ◽  
Mixing Sequences ◽  
Empirical Distribution Functions

We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}. Stationarity is not assumed. Here Fˆn is the perturbed empirical distribution function and Un is a U-statistic based on X1,…,Xn.

Sign in / Sign up

Export Citation Format

Share Document