Central limit theorems for double Poisson integrals

Details Central limit theorems for double Poisson integrals Central limit theorems for double Poisson integrals

2008-10-24

arxiv.org/abs/0810.4432v1

Bernoulli 2008, Vol. 14, No. 3, 791-821

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/08-BEJ123 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ123

Motivated by second order asymptotic results, we characterize the convergence in law of double integrals, with respect to Poisson random measures, toward a standard Gaussian distribution. Our conditions are expressed in terms of contractions of the kernels. To prove our main results, we use the theory of stable convergence of generalized stochastic integrals developed by Peccati and Taqqu. One of the advantages of our approach is that the conditions are expressed directly in terms of the kernel appearing in the multiple integral and do not make any explicit use of asymptotic dependence properties such as mixing. We illustrate our techniques by an application involving linear and quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as examples concerning random hazard rates.

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