The Annals of Probability

Free subexponentiality

Rajat Subhra Hazra and Krishanu Maulik

Full-text: Open access


In this article, we introduce the notion of free subexponentiality, which extends the notion of subexponentiality in the classical probability setup to the noncommutative probability spaces under freeness. We show that distributions with regularly varying tails belong to the class of free subexponential distributions. This also shows that the partial sums of free random elements having distributions with regularly varying tails are tail equivalent to their maximum in the sense of Ben Arous and Voiculescu [Ann. Probab. 34 (2006) 2037–2059]. The analysis is based on the asymptotic relationship between the tail of the distribution and the real and the imaginary parts of the remainder terms in Laurent series expansion of Cauchy transform, as well as the relationship between the remainder terms in Laurent series expansions of Cauchy and Voiculescu transforms, when the distribution has regularly varying tails.

Article information

Ann. Probab., Volume 41, Number 2 (2013), 961-988.

First available in Project Euclid: 8 March 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras
Secondary: 60G70: Extreme value theory; extremal processes

Free probability Cauchy transform Voiculescu transform regular variation convolution subexponential


Hazra, Rajat Subhra; Maulik, Krishanu. Free subexponentiality. Ann. Probab. 41 (2013), no. 2, 961--988. doi:10.1214/11-AOP706.

Export citation


  • [1] Banica, T., Curran, S. and Speicher, R. (2012). De Finetti theorems for easy quantum groups. Ann. Probab. 40 401–435.
  • [2] Ben Arous, G. and Kargin, V. (2010). Free point processes and free extreme values. Probab. Theory Related Fields 147 161–183.
  • [3] Ben Arous, G. and Voiculescu, D. V. (2006). Free extreme values. Ann. Probab. 34 2037–2059.
  • [4] Benaych-Georges, F. (2005). Failure of the Raikov theorem for free random variables. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 313–319. Springer, Berlin.
  • [5] Benaych-Georges, F. (2006). Taylor expansions of $R$-transforms: Application to supports and moments. Indiana Univ. Math. J. 55 465–481.
  • [6] Bercovici, H. and Pata, V. (1996). The law of large numbers for free identically distributed random variables. Ann. Probab. 24 453–465.
  • [7] Bercovici, H. and Pata, V. (1999). Stable laws and domains of attraction in free probability theory. Ann. of Math. (2) 149 1023–1060.
  • [8] Bercovici, H. and Pata, V. (2000). A free analogue of Hinčin’s characterization of infinite divisibility. Proc. Amer. Math. Soc. 128 1011–1015.
  • [9] Bercovici, H. and Pata, V. (2000). Functions of regular variation and freely stable laws. Ann. Mat. Pura Appl. (4) 178 245–269.
  • [10] Bercovici, H. and Voiculescu, D. (1993). Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 733–773.
  • [11] Bercovici, H. and Voiculescu, D. (1995). Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Theory Related Fields 103 215–222.
  • [12] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [14] Maassen, H. (1992). Addition of freely independent random variables. J. Funct. Anal. 106 409–438.
  • [15] Pata, V. (1996). The central limit theorem for free additive convolution. J. Funct. Anal. 140 359–380.
  • [16] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • [17] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. Wiley, Chichester.
  • [18] Speicher, R. (1994). Multiplicative functions on the lattice of noncrossing partitions and free convolution. Math. Ann. 298 611–628.
  • [19] Voiculescu, D. (1985). Symmetries of some reduced free product $C^{\ast}$-algebras. In Operator Algebras and Their Connections with Topology and Ergodic Theory (BuŞteni, 1983). Lecture Notes in Math. 1132 556–588. Springer, Berlin.
  • [20] Voiculescu, D. (1986). Addition of certain noncommuting random variables. J. Funct. Anal. 66 323–346.