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

\(p\)-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 \(\nu\).

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
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\textit{M. Grabchak}, J. Appl. Probab. 49, No. 4, 1015--1035 (2012; Zbl 1269.60018)

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