scholarly journals Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data

2016 ◽  
Vol 48 (2) ◽  
pp. 349-368
Author(s):  
Michael A. Kouritzin ◽  
Samira Sadeghi
Keyword(s):  
Long Range ◽  
Heavy Tails ◽  
Law Of Large Numbers ◽  
Nonlinear Function ◽  
Dependent Data ◽  
Long Range Dependence ◽  
Linear Processes ◽  
Strong Law ◽  
Large Numbers ◽  
Heavy Tailed

Abstract The Marcinkiewicz strong law, limn→∞(1 / n1/p)∑k=1n(Dk - D) = 0 almost surely with p ∈ (1, 2), is studied for outer products Dk = {XkX̅kT}, where {Xk} and {X̅k} are both two-sided (multivariate) linear processes (with coefficient matrices (Cl), (C̅l) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences Cl and C ̅l can decay slowly enough (as |l| → ∞) that {Xk,X ̅k} have long-range dependence, while {Dk} can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for {Dk} are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy tails or the long-range dependence, but not the combination. The main result is applied to obtain a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.

Asymptotics of the sample mean and sample covariance of long-range-dependent series

2004 ◽  
Vol 41 (A) ◽  
pp. 383-392
Author(s):  
Wen Dai
Keyword(s):  
Time Series ◽  
Long Range ◽  
Law Of Large Numbers ◽  
Stationary Time Series ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Sample Mean ◽  
Large Numbers ◽  
Sample Covariance ◽  
Dependent Series

An asymptotic distribution is given for the partial sums of a stationary time-series with long-range dependence. The law of large numbers for the sample covariance of the series is also derived. The results differ from those given elsewhere in relaxing the assumption of the independence of the innovations of the series.

scholarly journals Means as Improper Integrals

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 284
Author(s):  
John Gray ◽  
Andrew Vogt
Keyword(s):  
Law Of Large Numbers ◽  
Cauchy Distribution ◽  
Cumulative Distribution ◽  
Tail Probabilities ◽  
Strong Law ◽  
Improper Integral ◽  
Heavy Tailed Distributions ◽  
Large Numbers ◽  
The Mean ◽  
Heavy Tailed

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

Asymptotics of the sample mean and sample covariance of long-range-dependent series

2004 ◽  
Vol 41 (A) ◽  
pp. 383-392
Author(s):  
Wen Dai
Keyword(s):  
Time Series ◽  
Long Range ◽  
Law Of Large Numbers ◽  
Stationary Time Series ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Sample Mean ◽  
Large Numbers ◽  
Sample Covariance ◽  
Dependent Series

An asymptotic distribution is given for the partial sums of a stationary time-series with long-range dependence. The law of large numbers for the sample covariance of the series is also derived. The results differ from those given elsewhere in relaxing the assumption of the independence of the innovations of the series.

scholarly journals Means as Improper Integrals

2019 ◽  
Author(s):  
John E. Gray ◽  
Andrew Vogt
Keyword(s):  
Law Of Large Numbers ◽  
Cauchy Distribution ◽  
Cumulative Distribution ◽  
Tail Probabilities ◽  
Strong Law ◽  
Improper Integral ◽  
Heavy Tailed Distributions ◽  
Large Numbers ◽  
The Mean ◽  
Heavy Tailed

The aim of this work is to study generalizations of the notion of mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the Weak Law of Large Numbers in the same way that the ordinary mean relates to the Strong Law. We propose a further generalization, also based on an improper integral, called the doubly weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions, We also consider generalizations arising from Abel-Feynman type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

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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