Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 3, 763--819

A CONTEMPORARY REVIEW AND BIBLIOGRAPHY OF INFINITELY DIVISIBLE DISTRIBUTIONS AND PROCESSES

By

ARUP BOSE, Indian Statistical Institute, Kolkata

ANIRBAN DASGUPTA and HERMAN RUBIN

Purdue University

SUMMARY. This article provides a modern review of univariate and multivariate stable and infinitely divisible distributions and processes. Various characterizations and properties of stable and infinitely divisible distributions, including tail, moment and independence properties, and methods of simulation from an infinitely divisible distribution are discussed. Also discussed is the currently popular problem of estimating the index of a stable law and more generally, the heaviness of the tail of a distribution in the domain of attraction of a given stable law. A special feature of this article is its large collection of illustrative examples and a table of L\'evy measures.

AMS (1991) subject classification}. Primary 60E07; secondary 60-02, 62F10.

Key words and phrases. Infinitely divisible, stable, characteristic function, L\'evy measure, L\'evy process, operator stable, Poisson process, spectral measure, total variations, domain of attraction, compound Poisson, completely monotone, tail, moment, Hill estimate.

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