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Trees with random conductivities and the (reciprocal) inverse Gaussian distribution

Part of: Special processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  01 July 2016

O. E. Barndorff-Nielsen  and
A. E. Koudou
O. E. Barndorff-Nielsen*
Affiliation:
Aarhus University
A. E. Koudou*
Affiliation:
Université Paul Sabatier
*
∗ Postal address: Dept of Mathematical Sciences, Aarhus University, DK-8000 Aarhus C, Denmark. Email address: [email protected]
∗∗ Postal address: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, F-31062 Toulouse cedex, France.
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Abstract

Equipping the edges of a finite rooted tree with independent resistances that are inverse Gaussian for interior edges and reciprocal inverse Gaussian for terminal edges makes it possible, for suitable constellations of the parameters, to show that the total resistance is reciprocal inverse Gaussian (Barndorff-Nielsen 1994). This result is extended to infinite trees. Also, a connection to Brownian diffusion is established and, for the case of finite trees, an exact distributional and independence result is derived for the conditional model given the total resistance.

Keywords

Brownian diffusiongamma distributionhitting timesKirchhoff-Ohm laws

MSC classification

Primary: 60B99: None of the above, but in this section
Secondary: 60K99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 409 - 424
Copyright
Copyright © Applied Probability Trust 1998 

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