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

2019 ◽  
Vol 23 ◽  
pp. 922-946 ◽  
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.

On the optimality of stationary replacement strategies

1980 ◽  
Vol 17 (01) ◽  
pp. 178-186 ◽  
Author(s):  
Bo Bergman
Keyword(s):  
Large Class ◽  
Law Of Large Numbers ◽  
Martingale Differences ◽  
Long Run ◽  
Large Numbers ◽  
The Law

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.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-26 ◽  
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).

scholarly journals A strong law of large numbers for harmonizable isotropic random fields

1997 ◽  
Vol 10 (3) ◽  
pp. 219-225 ◽  
Author(s):  
Randall J. Swift
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Law Of Large Numbers ◽  
Natural Extension ◽  
Spherical Average ◽  
Strong Law ◽  
Isotropic Random Field ◽  
Large Numbers ◽  
Isotropic Random

The class of harmonizable fields is a natural extension of the class of stationary fields. This paper considers a strong law of large numbers for the spherical average of a harmonizable isotropic random field.

On the optimality of stationary replacement strategies

1980 ◽  
Vol 17 (1) ◽  
pp. 178-186 ◽  
Author(s):  
Bo Bergman
Keyword(s):  
Large Class ◽  
Law Of Large Numbers ◽  
Martingale Differences ◽  
Long Run ◽  
Large Numbers ◽  
The Law

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.

scholarly journals Strong Laws of Large Numbers for𝔹-Valued Random Fields

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Zbigniew A. Lagodowski
Keyword(s):  
Banach Space ◽  
Random Fields ◽  
Law Of Large Numbers ◽  
Random Vectors ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Geometry Of Banach Space ◽  
Necessary And Sufficient

We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for𝔹-valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.

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