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 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.

scholarly journals Variance asymptotics and central limit theorems for volumes of unions of random closed sets

2002 ◽  
Vol 34 (03) ◽  
pp. 520-539 ◽  
Author(s):  
Tomasz Schreiber
Keyword(s):  
Central Limit ◽  
Borel Measure ◽  
Strong Law ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Large Numbers ◽  
Multidimensional Ball ◽  
Mean Width ◽  
The Mean ◽  
Heavy Tailed

Let X, X 1, X 2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X 1 ∪ ∙ ∙ ∙ ∪ X n )) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝ d with centres distributed according to a spherically-symmetric heavy-tailed law.

scholarly journals Minimal waiting times in static traffic control

2003 ◽  
Vol 7 (1) ◽  
pp. 11-28
Author(s):  
O. Moeschlin ◽  
C. Poppinga
Keyword(s):  
Waiting Time ◽  
Traffic Control ◽  
Law Of Large Numbers ◽  
Waiting Times ◽  
Strong Law ◽  
Poisson Arrival ◽  
Large Numbers ◽  
Arrival Processes ◽  
The Mean ◽  
Traffic Organization

The paper discusses the question of the optimal control of an unsymmetric bottleneck system with Poisson arrival processes having the minimization of the mean individual waiting time as objective. The setup allows the straightforward generalization to more complicated forms of traffic organization. The notion of the mean individual waiting time is based on a theorem of the Little type, which is derived by a strong law of large numbers. The proof makes use of McNeil's formula, which connects the expected total waiting time with the expected queue length.

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (01) ◽  
pp. 182-209
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as thenearest neighbour graph ford,K≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (1) ◽  
pp. 182-209 ◽  
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the nearest neighbour graph for d, K ≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

scholarly journals Variance asymptotics and central limit theorems for volumes of unions of random closed sets

2002 ◽  
Vol 34 (3) ◽  
pp. 520-539 ◽  
Author(s):  
Tomasz Schreiber
Keyword(s):  
Central Limit ◽  
Borel Measure ◽  
Strong Law ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Large Numbers ◽  
Multidimensional Ball ◽  
Mean Width ◽  
The Mean ◽  
Heavy Tailed

Let X, X1, X2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X1 ∪ ∙ ∙ ∙ ∪ Xn)) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝd with centres distributed according to a spherically-symmetric heavy-tailed law.

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.

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