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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 28, Fasc. 1,
pages 21 - 40
 

CONVERGENCE IN VARIATION OF THE JOINT LAWS OF MULTIPLE STABLE STOCHASTIC INTEGRALS

Jean-Christophe Breton

Abstract: In this note, we are interested in the regularity in the sense of total variation of the joint laws of multiple stable stochastic integrals. Namely, we show that the convergence $$ (\cal L)\big (I_(d_1)(f_1^n), \dots , I_(d_p)(f_p^n)\big )\cvar (\cal L)\big (I_(d_1)(f_1), \dots , I_(d_p)(f_p)\big )\ \quad \text (as )n\to +\infty $$ holds true as long as each kernel $f_i^n$ converges when $n\negmedspace \to \negmedspace +\infty $ to $f_i$ in the Lorentz-type space $L^\alpha (\log _+)^(d_i-1)([0,1]^(d_i))$ for $1\leq i\leq p$. This result generalizes [4] from the one-dimensional case to the joint law case. It generalizes also [6] from the Wiener-It setting to the stable setting and [5] in the study of joint law of multiple stable integrals.

2000 AMS Mathematics Subject Classification: Primary: 60F99, 60G52, 60H05; Secondary: 60G57.

Key words and phrases: Convergence in variation, multiple stochastic integrals, stable process, LePage representation, method of superstructure.

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