The Annals of Probability

The Wiener condition and the conjectures of Embrechts and Goldie

Toshiro Watanabe

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show that the class of convolution equivalent distributions and the class of locally subexponential distributions are not closed under convolution roots. It gives a negative answer to the classical conjectures of Embrechts and Goldie. Moreover, we establish two sufficient conditions in order that the class of convolution equivalent distributions is closed under convolution roots.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1221-1239.

Dates
Received: February 2016
Revised: April 2017
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1556784018

Digital Object Identifier
doi:10.1214/17-AOP1195

Mathematical Reviews number (MathSciNet)
MR3945745

Zentralblatt MATH identifier
07067268

Subjects
Primary: 60E05: Distributions: general theory 60G50: Sums of independent random variables; random walks 62E20: Asymptotic distribution theory

Keywords
Convolution equivalence local subexponentiality convolution roots Wiener condition

Citation

Watanabe, Toshiro. The Wiener condition and the conjectures of Embrechts and Goldie. Ann. Probab. 47 (2019), no. 3, 1221--1239. doi:10.1214/17-AOP1195. https://projecteuclid.org/euclid.aop/1556784018


Export citation

References

  • [1] Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Probab. 16 489–518.
  • [2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [3] Borovkov, A. A. and Borovkov, K. A. (2008). Asymptotic Analysis of Random Walks. Heavy-Tailed Distributions. Encyclopedia of Mathematics and Its Applications 118. Cambridge Univ. Press, Cambridge. Translated from the Russian by O. B. Borovkova.
  • [4] Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Anal. Math. 26 255–302.
  • [5] Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Probab. 1 663–673.
  • [6] Čistjakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Teor. Veroyatn. Primen. 9 710–718.
  • [7] Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Probab. Theory Related Fields 72 529–557.
  • [8] Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Aust. Math. Soc. A 29 243–256.
  • [9] Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stochastic Process. Appl. 13 263–278.
  • [10] Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49 335–347.
  • [11] Foss, S. and Korshunov, D. (2007). Lower limits and equivalences for convolution tails. Ann. Probab. 35 366–383.
  • [12] Foss, S., Korshunov, D. and Zachary, S. (2013). An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd ed. Springer Series in Operations Research and Financial Engineering. Springer, New York.
  • [13] Korevaar, J. (2004). Tauberian Theory. A Century of Developments. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 329. Springer, Berlin.
  • [14] Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 407–424.
  • [15] Pakes, A. G. (2007). Convolution equivalence and infinite divisibility: Corrections and corollaries. J. Appl. Probab. 44 295–305.
  • [16] Rogozin, B. A. (1999). On the constant in the definition of subexponential distributions. Teor. Veroyatn. Primen. 44 455–458.
  • [17] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. Translated from the 1990 Japanese original, Revised by the author.
  • [18] Shimura, T. and Watanabe, T. (2005). Infinite divisibility and generalized subexponentiality. Bernoulli 11 445–469.
  • [19] Shimura, T. and Watanabe, T. (2005). On the convolution roots in the convolution-equivalent class. The Institute of Statistical Mathematics Cooperative Research Report 175, pp. 1–15.
  • [20] Teugels, J. L. (1975). The class of subexponential distributions. Ann. Probab. 3 1000–1011.
  • [21] Watanabe, T. (2008). Convolution equivalence and distributions of random sums. Probab. Theory Related Fields 142 367–397.
  • [22] Watanabe, T. and Yamamuro, K. (2010). Local subexponentiality and self-decomposability. J. Theoret. Probab. 23 1039–1067.
  • [23] Watanabe, T. and Yamamuro, K. (2010). Ratio of the tail of an infinitely divisible distribution on the line to that of its Lévy measure. Electron. J. Probab. 15 44–74.
  • [24] Watanabe, T. and Yamamuro, K. (2017). Two non-closure properties on the class of subexponential densities. J. Theoret. Probab. 30 1059–1075.
  • [25] Wiener, N. (1932). Tauberian theorems. Ann. of Math. (2) 33 1–100.
  • [26] Xu, H., Foss, S. and Wang, Y. (2015). Convolution and convolution-root properties of long-tailed distributions. Extremes 18 605–628.