The convergence of a sequence of random fields in the space \(D([0,1]^d)\). (English. Ukrainian original) Zbl 0998.60029

Kurchenko, O. O.

Theory Probab. Math. Stat. 64, 93-103 (2002); translation from Teor. Jmovirn. Mat. Stat. 64, 82-91 (2001).

This article presents the functional central limit theorem for a sequence of random fields constructed by properly normalized sums of nonlinear functions of Gaussian random fields with \(d\)-dimensional parameter \((d\geq 2)\). Convergence of the finite-dimensional distributions is proved via the method of moments while verifying the tightness conditions of measures (due to Bickel and Wichura) finishes the proof of weak convergence in the Skorokhod space. As a consequence certain limit theorems for the Baxter type sums for the Brownian sheet (\(d\)-parameter Brownian motion in the sense of Chentsov) and multiparameter fractional Brownian motion are obtained.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60F17	Functional limit theorems; invariance principles
60G60	Random fields

Keywords:

multiparameter Brownian sheet; multiparameter Brownian motion; Baxter type theorem; Baxter type sums; Hermite polynomials; Skorokhod space