Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 3, 868--883

DAMAGE MODELS --- A MARTIN BOUNDARY CONNECTION

By

C.R. RAO, Pennsylvania State University, USA, M. BHASKARA RAO, North Dakota State University, USA and D.N. SHANBHAG, University of Sheffield, UK

SUMMARY. This article is predominantly a review paper of the literature bringing Martin Boundary theory into the ambit of Damage Models. More specifically, it concerns the Martin Boundary in the environment of non-negative matrices with the inherent extreme point methods that is linked to Damage Models. Included in this paper are some new observations on certain results on damage models, which were obtained earlier following random walk and branching processes methods, amongst other things. De Finetti's theorem for exchangeable random variables has already been known to have links with certain results on the Integrated Cauchy Functional Equation (ICFE) (Shanbhag, 1977 and Lau and Rao, 1982). A special version of ICFE, or of de Finetti's theorem for discrete random variables plays a crucial role in the damage model studies. We bring the Martin boundary theory into the fold of damage model studies.

AMS (1991) subject classification}. 60G40, 91A60, 60E15, 46A55.

Key words and phrases. Choquet-Deny theorem, damage models, de Finetti's theorem, Deny's theorem, exchangeability, integrated Cauchy functional equation, ladder variables,

Lau-Rao-Shanbhag theorems, Martin boundary, Poisson-Martin representation

theorem, Perron-Frobenius theorem, random walk, Spitzer integral

representation.

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