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Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. (English) Zbl 1246.60039
A multilinear homogeneous sum of order \(d\) is any finite sum of products of this order of independent centered random variables. Two problem of major importance in probability and statistics are addressed in this paper: convergence of such sums towards a normal vector and towards a chi-squared one. The second moment is always assumed to be finite. Hence, a CLT type convergence suggests some limiting Wiener chaos. Moreover, in the case of convergence towards some normal vector, in this Wiener chaos, all coefficients except those of order one, vanish, while convergence to some chi-squared law signifies that in the limit, all coefficients vanish except those of order two. As shown in the Theorem 1.2, a CLT type convergence for Gaussian i.i.d. random variables is equivalent to a similar convergence of a sequence of multilinear homogeneous sums for any centered and independent random variables with a uniformly bounded third moment. This is called universality of Wiener chaos. A similar result holds true for chi-squared convergence. One of the versions of it, given in Theorem 1.12, states, in particular, that a sequence of multilinear homogeneous sums for centered Gaussian i.i.d. sequences converges in distribution to a chi-squared vector with a given degree of freedom if and only if, for any sequence of such sums, for any sequence of centered independent random variables, under Lyapunov’s moment condition, convergence in law to the same chi-squared vector holds. Multivariate versions of both results and some other equivalent statements are obtained.
Among the most important preceding papers on the subject are [P. de Jong, J. Multivariate Anal. 34, No. 2, 275–289 (1990; Zbl 0709.60019); E. Mossel, R. O’Donnell and K. Oleszkiewicz, Ann. Math. (2) 171, No. 1, 295–341 (2010; Zbl 1201.60031); I. Nourdin and G. Peccati, Ann. Probab. 37, No. 4, 1412–1426 (2009; Zbl 1171.60323); D. Nualart and G. Peccati, Ann. Probab. 33, No. 1, 177–193 (2005; Zbl 1097.60007); V. I. Rotar, J. Multivariate Anal. 9, 511–530 (1979; Zbl 0426.62013)].

MSC:
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
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