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Necessary conditions for characterization of laws via mixed sums

  • Anthony G. Pakes
Distribution

Abstract

SupposeX andY are independent and identically distributed, and independent ofU which satisfies 0≤U≤1. Recent work has centered on finding the lawsL(X) for whichXU(X+Y) where ℞ denotes equality in law. We show that this equation corresponds to a certain projective invariance property under random rotations. Implicitly or explicitly, it has been assumed that the characteristic function ofX has an expansion property near the origin. We show that solutions may be admitted in the absence of this condition when −logU has a lattice law. A continuous version of the basic problem replaces sums with a Lévy process. Instead we consider self-similar processes, showing that a solution exists only whenU is constant, and then all processes of a given order are admitted.

Key words and phrases

Distribution theory characterization semi-stable laws mixtures self-similar processes 

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Copyright information

© The Institute of Statistical Mathematics 1994

Authors and Affiliations

  • Anthony G. Pakes
    • 1
  1. 1.Department of MathematicsUniversity of Western AustraliaNedlandsAustralia

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