The Annals of Probability

Lower limits and equivalences for convolution tails

Serguei Foss and Dmitry Korshunov

Full-text: Open access


Suppose F is a distribution on the half-line [0,∞). We study the limits of the ratios of tails $\overline{F*F}(x)/\widebar F(x)$ as x→∞. We also discuss the classes of distributions ${\mathcal{S}}$, ${\mathcal{S}}(\gamma)$ and ${\mathcal{S}}^{*}$.

Article information

Ann. Probab., Volume 35, Number 1 (2007), 366-383.

First available in Project Euclid: 19 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory
Secondary: 60F10: Large deviations

Convolution tail convolution equivalency subexponential distribution


Foss, Serguei; Korshunov, Dmitry. Lower limits and equivalences for convolution tails. Ann. Probab. 35 (2007), no. 1, 366--383. doi:10.1214/009117906000000647.

Export citation


  • Chistyakov, V. P. (1964). A theorem on sums of independent random positive variables and its applications to branching processes. Theory Probab. Appl. 9 640--648.
  • Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Anal. Math. 26 255--302.
  • Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Probab. 1 663--673.
  • Cline, D. (1987). Convolutions of distributions with exponential and subexponential tailes. J. Aust. Math. Soc. 43 347--365.
  • Denisov, D., Foss, S. and Korshunov, D. (2004). Tail asymptotics for the supremum of a random walk when the mean is not finite. Queueing Syst. 46 15--33.
  • Embrechts, P. (1983). The asymptotic behaviour of series and power series with positive coefficients. Med. Konink. Acad. Wetensch. België 45 41--61.
  • Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stochastic Process. Appl. 13 263--278.
  • Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Probab. 25 132--141.
  • Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 407--424.
  • Rogozin, B. A. (2000). On the constant in the definition of subexponential distributions. Theory Probab. Appl. 44 409--412.
  • Rogozin, B. A. and Sgibnev, M. S. (1999). Strongly subexponential distributions and Banach algebras of measures. Siberian Math. J. 40 963--971.
  • Rudin, W. (1973). Limits of ratios of tails of measures. Ann. Probab. 1 982--994.
  • Teugels, J. L. (1975). The class of subexponential distributions. Ann. Probab. 3 1000--1011.