On total positivity of exchangeable random variables obtained by symmetrization, with applications to failure-dependent lifetimes

2019 ◽  
Vol 169 ◽  
pp. 95-109 ◽  
Author(s):  
E. Bezgina ◽  
M. Burkschat
Keyword(s):  
Random Variables ◽  
Total Positivity ◽  
Exchangeable Random Variables

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

scholarly journals On the Joint Distribution of Stopping Times and Stopped Sums in Multistate Exchangeable Trials

2014 ◽  
Vol 51 (2) ◽  
pp. 483-491 ◽  
Author(s):  
M. V. Boutsikas ◽  
D. L. Antzoulakos ◽  
A. C. Rakitzis
Keyword(s):  
Joint Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Independent Interest ◽  
Exchangeable Random Variables ◽  
Stopped Sums ◽  
Independent And Identically Distributed

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let ST,i be the stopped sum denoting the number of appearances of outcome 'i' in X1, …, XT, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, ST,0, ST,1, …, ST,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

Optimal couplings are totally positive and more

2004 ◽  
Vol 41 (A) ◽  
pp. 321-332 ◽  
Author(s):  
Paul Glasserman ◽  
David D. Yao
Keyword(s):  
Transportation Problem ◽  
Random Variables ◽  
Total Positivity ◽  
Bivariate Distribution ◽  
Discrete Distributions ◽  
Optimal Solutions ◽  
Totally Positive ◽  
Maximal Correlation ◽  
Optimal Coupling

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

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