Ordering Functions of Random Vectors, with Application to Partial Sums

2012 ◽  
Vol 26 (2) ◽  
pp. 474-479 ◽  
Author(s):  
Michel M. Denuit ◽  
Mhamed Mesfioui
Keyword(s):  
Partial Sums ◽  
Random Vectors

On Bernstein Type and Maximal Inequalities for Dependent Banach-Valued Random Vectors and Applications

2018 ◽  
Author(s):  
Noureddine Rhomari
Keyword(s):  
Recursive Estimation ◽  
Covariance Operator ◽  
Partial Sums ◽  
Strong Mixing ◽  
Random Vectors ◽  
Bernstein Type ◽  
Strong Law ◽  
Maximal Inequalities ◽  
Mixing Coefficients ◽  
Measure Of Dependence

This article discusses some results on Bernstein type and maximal inequalities for partial sums of dependent random vectors taking their values in separable Hilbert or Banach spaces of finite or infinite dimension. Two types of measure of dependence are considered: strong mixing coefficients (α-mixing) and absolutely regular mixing coefficients (β-mixing). These inequalities, which are similar to those in the dependent real case, are used to derive the strong law of large numbers (SLLN) and the bounded law of the iterated logarithm (LIL) for absolutely regular Hilbert- or Banach-valued processes under minimal mixing conditions. The article first introduces the relevant notation and definitions before presenting the maximal inequalities in the strong mixing case, followed by the absolutely regular mixing case. It concludes with some applications to the SLLN, the bounded LIL for Hilbertian or Banachian absolutely regular processes, the recursive estimation of probability density, and the covariance operator estimations.

On Maxima and Minima of Partial Sums of Strongly Interchangeable Random Variables

1990 ◽  
Vol 4 (3) ◽  
pp. 319-332 ◽  
Author(s):  
Teunis J. Ott ◽  
J. George Shanthikumar
Keyword(s):  
Random Vector ◽  
Queueing Networks ◽  
Joint Distribution ◽  
Random Variables ◽  
Markov Property ◽  
Partial Sums ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Voice Communication ◽  
Special Instance

We introduce the concept of “strong interchangeability” of random vectors. Strongly interchangeable random vectors arise naturally in packetized voice channels, M/G/1 queues, symmetric queueing networks, and other standard symmetric distributions. We study some properties of strongly interchangeable random vectors. We show that if (X1, …, XN) is a strongly interchangeable random vector, then even though there is no Markov property, taboo probabilities can be used to compute the joint distribution of ŽN = min1≤n≤N σnk=IXk and ZN = max1≤n≤N σnk=1Xk. For a special instance of this problem that arises in packetized voice communication, it is shown that the resulting algorithm essentially has a complexity of order N4. When ( σnk=1Xk, n = 1,… N) is an associated random vector bound for the joint distribution of ŽN and ZN are obtained and applied to the packetized voice communication problem.

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