## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 4 (1994), 2242-2274.

### Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors

#### Abstract

Limit theorems for functions of stationary mean-zero Gaussian sequences of vectors satisfying long range dependence conditions are considered. Depending on the rate of decay of the coefficients, the limit law can be either Gaussian or the law of a multiple Ito-Wiener integral. We prove the bootstrap of these limit theorems in the case when the limit is normal. A sufficient bracketing condition for these limit theorems to happen uniformly over a class of functions is presented.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 4 (1994), 2242-2274.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988503

**Digital Object Identifier**

doi:10.1214/aop/1176988503

**Mathematical Reviews number (MathSciNet)**

MR1331224

**Zentralblatt MATH identifier**

0839.60024

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60F17: Functional limit theorems; invariance principles 60G10: Stationary processes

**Keywords**

Long range dependence bootstrap multiple Ito-Wiener integrals moving blocks bootstrap empirical processes stationary Gaussian sequence

#### Citation

Arcones, Miguel A. Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors. Ann. Probab. 22 (1994), no. 4, 2242--2274. doi:10.1214/aop/1176988503. https://projecteuclid.org/euclid.aop/1176988503