Convergence in variation of the laws of multiple stable integrals. (Q1426590)

The author extends a result of \textit{Yu. A. Davydov} and \textit{G. V. Martynova} [in: Statistics and control of stochastic processes, 55--57 (1989; Zbl 0709.60056)] on the convergence in variation of the laws of multiple Wiener integrals to multiple stable integrals. Namely, he proves that if a sequence \((f_n)_{n>0}\) converges to \(f\) in \(L^\alpha (\log_+ )^{d-1} ([0,1]^d )\), then the law of \(I_d (f_n)\) converges in variation to the law of \(I_d(f)\), where \(I_d(f)\) denotes the the multiple stable integral of \(f\). The proof uses the LePage representation of random polynomials obtained by conditioning multiple stable integrals.