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Multivariate extremal processes, leader processes and dynamic choice models

Published online by Cambridge University Press:  01 July 2016

Sidney Resnick  and
Rishin Roy
Sidney Resnick*
Affiliation:
Cornell University
Rishin Roy*
Affiliation:
Cornell University
*
∗Postal address: Cornell University, School of OR & IE, Upson Hall, Ithaca, NY 14853, USA.
∗∗Postal address: Cornell University, Johnson Graduate School of Management, Malott Hall, Ithaca, NY 14853, USA.
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Abstract

Let (Y(t), t > 0) be a d-dimensional non-homogeneous multivariate extremal process. We suppose the ith component of Y describes time-dependent behaviour of random utilities associated with the ith choice. At time t we choose the ith alternative if the ith component of Y(t) is the largest of all the components. Let J(t) be the index of the largest component at time t so J has range {1, …, d} and call {J(t)} the leader process. Let Z(t) be the value of the largest component at time t. Then the bivariate process (J(t), Z(t)} is Markov. We discuss when J(t) and Z(t) are independent, when {J(s), 0<s≦t} and Z(t) are independent and when J(t) and {Z(s), 0<s≦t} are independent. In usual circumstances, {J(t)} is Markov and particular properties are given when the underlying distribution is max-stable. In the max-stable time-homogeneous case, {J(et)} is a stationary Markov chain with stationary transition probabilities.

Keywords

UTILITYMARKOV CHAIN
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 2 , June 1990 , pp. 309 - 331
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

Partly supported by NSF Grant MCS-8801034 and the Mathematical Sciences Institute at Cornell University.

Supported by the Johnson Graduate School of Management, Cornell University.

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