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ON A POISSON HYPERBOLIC STAIRCASE

Published online by Cambridge University Press:  01 January 1999

Benny Levikson ,
Tomasz Rolski  and
Gideon Weiss
Benny Levikson
Affiliation:
Department of Statistics, Haifa University, Haifa, Israel
Tomasz Rolski
Affiliation:
Mathematical Institute, Wrocław University, Wrocław, Poland
Gideon Weiss
Affiliation:
Department of Statistics, Haifa University, Haifa, Israel
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Abstract

We consider a process that starts at height y, stays there for a time X0 ∼ exp(y) when it drops to a level Z1 ∼ U(0, y). Thereafter it stays at level Zn for time Xn ∼ exp(Zn), then drops to a level Zn+1 ∼ U(0,Zn). We investigate properties of this process, as well as the Poisson hyperbolic process which is obtained by randomizing the starting point y of the above process. This process is associated with a rate 1 Poisson process in the positive quadrant: Its path is the minimal RCLL decreasing step function through Poisson points in the positive quadrant. The finite dimensional distributions are then multivariate exponential in sense of Marshall-Olkin.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 13 , Issue 1 , January 1999 , pp. 11 - 31
Copyright
© 1999 Cambridge University Press

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