Some New Approaches to Infinite Divisibility

Theofanis Sapatinas (University of Cyprus)
Damodar Shanbhag (University of Cyprus)
Arjun K Gupta (Bowling Green State University)


Using an approach based, amongst other things, on Proposition 1 of Kaluza (1928), Goldie (1967) and, using a different approach based especially on zeros of polynomials, Steutel (1967) have proved that each nondegenerate distribution function (d.f.) $F$ (on $\mathbb{R}$, the real line), satisfying $F(0-)=0$ and $F(x)=F(0)+(1-F(0))G(x), x > 0$, where $G$ is the d.f. corresponding to a mixture of exponential distributions, is infinitely divisible. Indeed, Proposition 1 of Kaluza (1928) implies that any nondegenerate discrete probability distribution $\{p_x:x=0,1,\ldots\}$ that is log-convex or, in particular, completely monotone, is compound geometric, and, hence, infinitely divisible. Steutel (1970), Shanbhag & Sreehari (1977) and Steutel & van Harn (2004, Chapter VI) have given certain extensions or variations of one or more of these results. Following a modified version of the C.R. Rao et al. (2009, Section 4) approach based on the Wiener-Hopf factorization, we establish some further results of significance to the literature on infinite divisibility.

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Pages: 2359-2374

Publication Date: November 21, 2011

DOI: 10.1214/EJP.v16-961


  1. Blum, J.R. and Rosenblatt, M. (1959). On the structure of infinitely divisible distributions. Pacific J. Math., 9, 1-7. Math. Review 105729.
  2. Donoghue, W.F., Jr. (1969). Distributions and Fourier Transforms. New York: Academic Press. Math. Review number not available.
  3. Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. II, 2nd Edition, New York: John Wiley & Sons. Math. Review 270403.
  4. Goldie, C.M. (1967). A class of infinitely divisible random variables. Proc. Cambridge Philos. Soc., 63, 1141-1143. Math. Review 215332.
  5. Hardy, G., Littlewood, J.E. & Pólya, G. (1952). Inequalities. 2nd Edition, Cambridge University Press, Cambridge. Math. Review 46395.
  6. Kaluza, T. (1928). "Uber die koeffizienten reziproker potenzreihen. Math. Z., 28, 161-170. Math. Review 154494.
  7. Kingman, J.F.C. (1972). Regenerative Phenomena. New York: John Wiley & Sons. Math. Review 350861.
  8. Loève, M. (1963). Probability Theory. 3rd Edition, Princeton: Van Nostrand. Math. Review 203748.
  9. Lukacs, E. (1970). Characteristic Functions. 2nd Edition, London: Griffin. Math. Review 3468748.
  10. Rao, C.R. & Shanbhag, D.N. (1994). Choquet-Deny Type Functional Equations with Applications to Stochastic Models. Chichester: John Wiley & Sons. Math. Review 1329995.
  11. Rao, C.R., Shanbhag, D.N., Sapatinas, T. & Rao, M.B. (2009). Some properties of extreme stable laws and related infinitely divisible random variables. J. Statist. Plann. Inference, 139, 802-813. Math. Review 2479829.
  12. Shanbhag, D.N. (1977). On renewal sequences. Bull. London Math. Soc., 9, 79-80. Math. Review 428497.
  13. Shanbhag, D.N. & Sreehari, M. (1977). On certain self-decomposable distributions. Z. Wahrsch. Verw. Gebiete, 38, 217-222. Math. Review 436267.
  14. Shanbhag, D.N., Pestana, D. & Sreehari, M. (1977). Some further results in infinite divisibility. Math. Proc. Cambridge Philos. Soc., 82, 289-295. Math. Review 448483.
  15. Steutel, F.W. (1967). Note on the infinite divisibility of exponential mixtures. Ann. Math. Statist., 38, 1303-1305. Math. Review 215339.
  16. Steutel, F.W. (1970). Preservation of Infinite Divisibility under Mixing and Related Topics. Mathematical Centre Tracts, Vol. 33, Amsterdam: Mathematisch Centrum. Math. Review 278355.
  17. Steutel, F.W. & van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. New York: Marcel Dekker. Math. Review 2011862.
  18. Titchmarsh, E.C. (1978). The Theory of Functions. 2nd Edition, Oxford: Oxford University Press. Math. Review number not available.
  19. Zygmund, A. (2002). Trigonometric Series. Volumes I & II, 3rd Edition. Cambridge: Cambridge University Press. Math. Review 1963498.

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