On a new class of tempered stable distributions: moments and regular variation. (English) Zbl 1269.60018

Grabchak, Michael

J. Appl. Probab. 49, No. 4, 1015-1035 (2012).

\(p\)-tempered \(\alpha\)-stable distributions on \(\mathbb{R}^d\) are infinitely divisible laws \(\mu\) with Lévy measure \(B\mapsto M(B):= \int_{\mathbb{R}^d} 1_B(r u)q(r^p,u) r^{-\alpha-1} dr d\sigma(u)\) (\(B\) denoting a Borel set in \(\mathbb{R}^d \backslash\{0\}\))), where \(\sigma \in M^b_+(S^{d-1})\) and \(q: \mathbb{R}_+^\times \times S^{d-1}\to\mathbb{R}_+^\times\) is a Borel function such that, for all \(u\), \(r\mapsto q(r,u)\) is completely monotone with \(\lim_{r\to\infty}q(r,u)=0\). Furthermore, it is supposed that \(\alpha < 2\), \(p>0\) and \(\mu\) has no Gaussian component. \(\mu\) is called proper if \(\lim_{r\to 0}q(r,u)=1\) for all \(u\). If \(0<\alpha<2\) and \(p=1\), the class coincides with tempered \(\alpha\)-stable distributions investigated by J. Rosiński [Stochastic Processes Appl. 117, No. 6, 677–707 (2007; Zbl 1118.60037)], more general, for \(p>0\), cf. by J. Rosiński and J. Sinclair [Banach Center Publications 90, 153-170 (2010; Zbl 1210.60048)]. In this case, \(M\) is absolutely continuous with respect to the Lévy measure of a stable distribution.
The author generalizes the concept of Rosiński (and various others), admitting also negative values \(\alpha\). The absolute monotone functions \(q(\cdot,u)\) define (via Bernstein’s theorem) a family \((Q_u)\) of Borel measures and a re-parametrization (defined in [Rosiński, loc. cit.]) yields a unique measure \(R\) (called here Rosiński measure). \((Q_u)\), \(\sigma\) and the Lévy measure \(M\) are determined by \(R\) (Theorem 1). Hence, the notation \(\mu\in TS_\alpha^p(R,b)\) is justified, where \(b\) denotes the drift term in the Lévy-Khinchin representation of \(\mu\).
For non-proper \(p\)-tempered \(\alpha\)-stable laws \(\mu\), the parameters \(R\) and \(\alpha\) are in general not identifiable. In particular, for \(0<\alpha<2\) and \(\beta \in (\alpha, 2)\), \(\mu\in TS_\beta^p(R,b)\) is representable as \(\mu\in TS_\alpha^p(R',b)\) (for a Rosiński measure \(R'\)).
Section 3 is concerned with the existence of moments and exponential moments of \(\mu\) and Section 4 with regular variation of the tail of a Rosiński measure \(R\), thus determining \(\gamma\)-stable laws \(\nu\) such that \(\mu\in TS_\alpha^p(R,b)\) belongs to the domain of attraction of \(\nu\).

Reviewer: Wilfried Hazod (Dortmund)

MSC:

60E07	Infinitely divisible distributions; stable distributions
60G51	Processes with independent increments; Lévy processes

Keywords:

infinite divisible law; tempered stable distribution; regular variation; tail behaviour; domain of attraction; Rosiński measure

Citations:

Zbl 1118.60037; Zbl 1210.60048

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