On the optimality of stationary replacement strategies

In this paper it is shown that for a large class of replacement problems the class of stationary replacement strategies is complete, i.e. in order to minimize the average long run cost per unit time it suffices to consider replacement rules which are equal for each new unit irrespectively of what has been observed from earlier units. The main result is based on a version of the law of large numbers for martingale differences proved in the appendix.

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Deviation inequalities for Banach space valued martingales differences sequences and random fields

ESAIM Probability and Statistics ◽

10.1051/ps/2019016 ◽

2019 ◽

Vol 23 ◽

pp. 922-946 ◽

Cited By ~ 1

Author(s):

Davide Giraudo

Keyword(s):

Banach Space ◽

Linear Regression ◽

Random Field ◽

Random Fields ◽

Law Of Large Numbers ◽

Partial Sums ◽

Martingale Differences ◽

Large Numbers ◽

Deviation Inequalities ◽

The Law

We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of an orthomartingale differences random field. These inequalities can be used to give rates for linear regression and the law of large numbers.

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Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

Abstract and Applied Analysis ◽

10.1155/2013/715054 ◽

2013 ◽

Vol 2013 ◽

pp. 1-26 ◽

Cited By ~ 1

Author(s):

Shunli Hao

Keyword(s):

Law Of Large Numbers ◽

Convergence Rates ◽

Sufficient Conditions ◽

Random Variables ◽

General Setting ◽

Weighted Sums ◽

Moment Condition ◽

Martingale Differences ◽

Large Numbers ◽

The Law

We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), Spitzer (1956), and Baum and Katz (1965). In the real valued single martingale case, it generalizes the results of Alsmeyer (1990). The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinite many Banach valued martingale differences. The obtained results improve and extend those of Ghosal and Chandra (1998).

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Convergence rates in the law of large numbers for arrays of martingale differences

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2014.03.049 ◽

2014 ◽

Vol 417 (2) ◽

pp. 733-773 ◽

Cited By ~ 3

Author(s):

Shunli Hao ◽

Quansheng Liu

Keyword(s):

Law Of Large Numbers ◽

Convergence Rates ◽

Martingale Differences ◽

Large Numbers ◽

The Law

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The Roll of the Dice

10.1093/oso/9780198869023.003.0001 ◽

2021 ◽

pp. 11-40

Author(s):

Andrew C. A. Elliott

Keyword(s):

Law Of Large Numbers ◽

Gambler’S Fallacy ◽

Long Run ◽

Games Of Chance ◽

Large Numbers ◽

The Asymmetry ◽

Gambler's Fallacy ◽

The Law

The topic of probability is introduced through analysis of games of chance, using the casino games of roulette and the dice game craps. The nature of probability is explored, including different interpretations of what probability actually is. Ways of combining probabilities are described. The player will lose in the long run, but how long a run is needed for this to show itself? The asymmetry between the player and the gambling house is explored. The Gambler’s Fallacy is contrasted to the law of large numbers.

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The y-Increment Puzzle, the Law-of-Large-Numbers, and a Second New Proof of Fermat’s Last Conjecture.

SSRN Electronic Journal ◽

10.2139/ssrn.3583363 ◽

2020 ◽

Cited By ~ 1

Author(s):

Michael C. I. Nwogugu

Keyword(s):

Law Of Large Numbers ◽

Large Numbers ◽

The Law

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A new proof for the generalized law of large numbers under Choquet expectation

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02426-5 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Jing Chen ◽

Zengjing Chen

Keyword(s):

Probability Theory ◽

Law Of Large Numbers ◽

Moment Condition ◽

Large Numbers ◽

Choquet Expectation ◽

The Law ◽

Linear Probability ◽

First Moment ◽

Independent And Identically Distributed

Abstract In this article, we employ the elementary inequalities arising from the sub-linearity of Choquet expectation to give a new proof for the generalized law of large numbers under Choquet expectations induced by 2-alternating capacities with mild assumptions. This generalizes the Linderberg–Feller methodology for linear probability theory to Choquet expectation framework and extends the law of large numbers under Choquet expectation from the strong independent and identically distributed (iid) assumptions to the convolutional independence combined with the strengthened first moment condition.

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