Multivariate extremal processes, leader processes and dynamic choice models

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 i th 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 ( e t )} is a stationary Markov chain with stationary transition probabilities.