Summary: Statistics & Probability Letters 76 (2006) 20372042
Product formula and independence criterion for multiple
Se´ bastien Chivoret, Anna AmirdjanovaÃ,1
Department of Statistics, University of Michigan, 439 West Hall, 1085 S. University Avenue, Ann Arbor, MI 48109, USA
Received 6 October 2005; received in revised form 29 November 2005
Available online 28 August 2006
Multiple stochastic integrals of Huang and Cambanis [1978. Stochastic and multiple Wiener integrals for Gaussian
processes. Ann. Probab. 6, 585614] with respect to a general Gaussian process X ¼ ðXt; t 2 TÞ, whose covariance function
is of bounded variation on bounded subsets of T Â T, are considered. A product formula for the integrals is derived and a
necessary and sufficient condition for independence of multiple HuangCambanis integrals is obtained. As an illustration,
the results are applied to the special case of multiple integrals with respect to a persistent fractional Brownian motion.
r 2006 Elsevier B.V. All rights reserved.
MSC: 60G15; 60H05
Keywords: Gaussian process; Multiple stochastic integral; Product formula; Independence; Fractional Brownian motion
The problem of finding necessary and sufficient conditions for independence of multiple stochastic integrals
in terms of their deterministic integrands was initially addressed by U¨ stu¨ nel and Zakai (1989) in the context of