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Hazra, Rajat Subhra; Maulik, Krishanu

Free subexponentiality. (English) Zbl 1276.46054

Ann. Probab. 41, No. 2, 961-988 (2013).
Summary: We introduce the notion of free subexponentiality, which extends the notion of subexponentiality in the classical probability setup to 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 G. Ben Arous and D. V. Voiculescu [Ann. Probab. 34, No. 5, 2037–2059 (2006; Zbl 1117.46044)]. 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.

Cited in 5 Documents

MSC:

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

Keywords:

free probability; Cauchy transform; Voiculescu transform; regular variation; convolution; subexponential

Citations:

Zbl 1117.46044
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References:

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