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The theory of Wiener-Itô integrals in vector-valued Gaussian stationary random fields. II. (English) Zbl 07743691

Summary: This work is the continuation of my paper in [Mosc. Math. J. 20, No. 4, 749–812 (2020; Zbl 1471.60046)]. In that paper the existence of the spectral measure of a vector-valued stationary Gaussian random field is proved and the vector-valued random spectral measure corresponding to this spectral measure is constructed. The most important properties of this random spectral measure are formulated, and they enable us to define multiple Wiener-Itô integrals with respect to it. Then an important identity about the products of multiple Wiener-Itô integrals, called the diagram formula is proved. In this paper an important consequence of this result, the multivariate version of Itô’s formula is presented. It shows a relation between multiple Wiener-Itô integrals with respect to vector-valued random spectral measures and Wick polynomials. Wick polynomials are the multivariate versions of Hermite polynomials. With the help of Itô’s formula the shift transforms of a random variable given in the form of a multiple Wiener-Itô integral can be written in a useful form. This representation of the shift transforms makes possible to rewrite certain non-linear functionals of a vector-valued stationary Gaussian random field in such a form which suggests a limiting procedure that leads to new limit theorems. Finally, this paper contains a result about the problem when this limiting procedure may be carried out, i.e., when the limit theorems suggested by our representation of the investigated non-linear functionals are valid.

MSC:

60G10 Stationary stochastic processes
60G15 Gaussian processes
60F99 Limit theorems in probability theory

Citations:

Zbl 1471.60046
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References:

[1] M. A. Arcones, Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors, Ann. Probab. 22 (1994), no. 4, 2242-2274. MR 1331224 · Zbl 0839.60024
[2] P. Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396 · Zbl 0172.21201
[3] P. Breuer and P. Major, Central limit theorems for nonlinear functionals of Gaussian fields, J. Multivariate Anal. 13 (1983), no. 3, 425-441. MR 716933 · Zbl 0518.60023
[4] R. L. Dobrushin, Gaussian and their subordinated self-similar random generalized fields, Ann. Probab. 7 (1979), no. 1, 1-28. MR 515810 · Zbl 0392.60039
[5] R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gauss-ian fields, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27-52. MR 550122 · Zbl 0397.60034
[6] B. Hajek and E. Wong, Multiple stochastic integrals: projection and iteration, Z. Wahrsch. Verw. Gebiete 63 (1983), no. 3, 349-368. MR 705629 · Zbl 0496.60049
[7] K. Itô, Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), 157-169. MR 44064 · Zbl 0044.12202
[8] P. Major, Multiple Wiener-Itô integrals, with applications to limit theorems, second ed., Lecture Notes in Mathematics, vol. 849, Springer, Cham, 2014. MR 3155040 · Zbl 1301.60004
[9] P. Major, Limit theorems for non-linear functionals of stationary Gaussian random fields, preprint arXiv:1708.03313 [math.PR]. · Zbl 0397.60034
[10] P. Major, The theory of Wiener-Itô integrals in vector valued Gaussian stationary random fields. Part I, Mosc. Math. J. 20 (2020), no. 4, 749-812. MR 4203058 · Zbl 1471.60046
[11] P. Major, Non-central limit theorem for non-linear functionals of vector-valued stationary gaussian random fields, To be submitted to the Annals of Probability, 2021.
[12] I. Nourdin and G. Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012. MR 2962301. From Stein’s method to universality. · Zbl 1266.60001
[13] M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 53-83. MR 550123 · Zbl 0397.60028
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