Strong asymptotic independence on Wiener chaos

Nourdin, Ivan; Nualart, David; Peccati, Giovanni

doi:10.1090/proc12769

Strong asymptotic independence on Wiener chaos
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by Ivan Nourdin, David Nualart and Giovanni Peccati PDF

Proc. Amer. Math. Soc. 144 (2016), 875-886 Request permission

Abstract:

Let $F_n = (F_{1,n}, \dots ,F_{d,n})$, $n\geqslant 1$, be a sequence of random vectors such that, for every $j=1,\dots ,d$, the random variable $F_{j,n}$ belongs to a fixed Wiener chaos of a Gaussian field. We show that, as $n\to \infty$, the components of $F_n$ are asymptotically independent if and only if $\mathrm {Cov}(F_{i,n}^2,F_{j,n}^2)\to 0$ for every $i\neq j$. Our findings are based on a novel inequality for vectors of multiple Wiener-Itô integrals, and represent a substantial refining of criteria for asymptotic independence in the sense of moments recently established by Nourdin and Rosiński (2014).

References

Shuyang Bai and Murad S. Taqqu, Multivariate limit theorems in the context of long-range dependence, J. Time Series Anal. 34 (2013), no. 6, 717–743. MR 3127215, DOI 10.1111/jtsa.12046
Solesne Bourguin and Jean-Christophe Breton, Asymptotic Cramér type decomposition for Wiener and Wigner integrals, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013), no. 1, 1350005, 13. MR 3071457, DOI 10.1142/S0219025713500057
Rumen L. Mishkov, Generalization of the formula of Faa di Bruno for a composite function with a vector argument, Int. J. Math. Math. Sci. 24 (2000), no. 7, 481–491. MR 1781515, DOI 10.1155/S0161171200002970
Ivan Nourdin, Lectures on Gaussian approximations with Malliavin calculus, Séminaire de Probabilités XLV, Lecture Notes in Math., vol. 2078, Springer, Cham, 2013, pp. 3–89. MR 3185909, DOI 10.1007/978-3-319-00321-4_{1}
Ivan Nourdin, A webpage on Stein’s method and Malliavin calculus, https://sites.google.com/site/malliavinstein
Ivan Nourdin and David Nualart, Central limit theorems for multiple Skorokhod integrals, J. Theoret. Probab. 23 (2010), no. 1, 39–64. MR 2591903, DOI 10.1007/s10959-009-0258-y
Ivan Nourdin and Giovanni Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012. From Stein’s method to universality. MR 2962301, DOI 10.1017/CBO9781139084659
Ivan Nourdin and Giovanni Peccati (2013): The optimal fourth moment theorem. To appear in: Proceedings of the American Mathematical Society.
Ivan Nourdin and Jan Rosiński, Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws, Ann. Probab. 42 (2014), no. 2, 497–526. MR 3178465, DOI 10.1214/12-AOP826
David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
David Nualart and Giovanni Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005), no. 1, 177–193. MR 2118863, DOI 10.1214/009117904000000621
D. Nualart and M. Zakai, A summary of some identities of the Malliavin calculus, Stochastic partial differential equations and applications, II (Trento, 1988) Lecture Notes in Math., vol. 1390, Springer, Berlin, 1989, pp. 192–196. MR 1019603, DOI 10.1007/BFb0083946
Giovanni Peccati and Ciprian A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, Séminaire de Probabilités XXXVIII, Lecture Notes in Math., vol. 1857, Springer, Berlin, 2005, pp. 247–262. MR 2126978, DOI 10.1007/978-3-540-31449-3_{1}7
Eric V. Slud, The moment problem for polynomial forms in normal random variables, Ann. Probab. 21 (1993), no. 4, 2200–2214. MR 1245307
Ali Süleyman Üstünel and Moshe Zakai, On independence and conditioning on Wiener space, Ann. Probab. 17 (1989), no. 4, 1441–1453. MR 1048936