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 whichX ℞U(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.
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References
Alamatsaz, M. H. (1993). On characterizations of exponential and gamma distributions,Statist. Probab. Lett.,17, 315–319.
Alzaid, A. A. and Al-Osh, M. A. (1991). Characterization of probability distributions based on the relationX ℞U (X 1 +X 2),Sankhyā Ser. B,53, 188–190.
Chang, D. K. (1989). A note on a characterization of gamma distributions,Utilitas Math.,35, 153–154.
Cowan, R. (1980). Problem corner,D.M.S. Newsletter, Nos. 65 and 66.
Kagan, A. M., Linnik, Yu V. and Rao, C. R. (1973).Characterization Theorems in Mathematical Statistics, Wiley, New York.
Lamperti, J. W. (1962). Semi-stable stochastic processes,Trans. Amer. Math. Soc.,104, 62–78.
Maejima, M. (1989). Self-similar processes and limit theorems,Sugaku Expositions,2, 103–123.
Pakes, A. G. (1992a). A characterization of gamma mixtures of stable laws motivated by limit theorems,Statistica Neerlandica, 209–218.
Pakes, A. G. (1992b). On characterizations via mixed sums,Austral. J. Statist.,34(2), 323–339.
Pakes, A. G. (1993). Characterization of discrete laws via mixed sums and Markov branching processes,Stochastic Process. Appl. (to appear).
Ramachandran, B. and Lau, K. S. (1991).Functional Equations in Probability Theory, Academic Press, New York.
Shanbhag, D. N. (1972). Characterizations under unimodality for exponential distribution, Research Report 99/DNS, Department of Probability and Statistics, The University of Sheffield, U.K.
van Harn, K. and Steutel, F. W. (1993). Stability equations for processes with stationary independent increments using branching processes and Poisson mixtures,Stochastic Process. Appl.,45, 209–230.
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This research was in part supported by NSERC grant A-8466.
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Pakes, A.G. Necessary conditions for characterization of laws via mixed sums. Ann Inst Stat Math 46, 797–802 (1994). https://doi.org/10.1007/BF00773483
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DOI: https://doi.org/10.1007/BF00773483