IJPAM: Volume 55, No. 2 (2009)

PROBABILITIES ON SIMPLY CONNECTED NILPOTENT
LIE GROUPS: ON THE DOEBLIN-GNEDENKO CONDITIONS
FOR THE DOMAIN OF ATTRACTION OF STABLE LAWS.
WITH AN APPENDIX ON A NEW PROOF OF SIEBERT'S
CONVERGENCE THEOREM FOR GENERATING
DISTRIBUTIONS

Daniel NeuenschwanderDepartment of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, Bern, CH-3012, SWITZERLAND
Department of Mathematical Statistics and Actuarial Science
University of Bern
Sidlerstrasse 5, Bern, CH-3012, SWITZERLAND
École des Hautes Etudes Commerciales
Institut de Sciences Actuarielles
Université de Lausanne
Lausanne, CH-1015, SWITZERLAND
Institut de Science Financi ${\rm\grave{e}}$re et d'Assurances
Université de Lyon
Université Claude Bernard Lyon 1
50, Avenue Tony Garnier, Lyon, F-69007, FRANCE
e-mail: daniel.neuenschwander@bluewin.ch


Abstract.The classical Doeblin-Gnedenko conditions characterizing the domain of attraction of a non-Gaussian stable law have been proved to be sufficient for completely non-Gaussian stable continuous convolution semigroups (c.c.s. for short) on simply connected nilpotent Lie groups by Carnal [#!car86!#]. His method was a translation of the extreme-value-theoretic approach due to Le Page, Woodroofe, Zinn [#!lep:woo:zin81!#]. In the present note, we give a generalization of the classical proof for the sufficiency of these conditions. Together with Neuenschwander [#!neu95a!#] this yields the fact that in case every eigenvalue of the square matrix $A$ has real part strictly greater than $1/2$, then for a completely non-Gaussian
$\{t^{A}\}_{t> 0}$-stable (in the not necessarily strict sense) semigroup on simply connected nilpotent Lie groups, the analogue of the Doeblin-Gnedenko conditions in fact characterize the non-strict $\{t^{A}\}_{t> 0}$-domain of attraction. We take this opportunity to state a new proof of Siebert's Convergence Theorem for c.c.s. and their generating distributions for the special case of simply connected nilpotent Lie groups and related results.

Received: July 6, 2009

AMS Subject Classification: 22E25, 60B15, 60F05

Key Words and Phrases: stable semigroups, domains of attraction, nilpotent Lie groups, convergence of generating distributions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 2