## A Limit Theorem for Weyl Transformation in Infinite-Dimensional Torus and Central Limit Theorem for Correlated Multiple Wiener Integrals

J. Math. Sci. Univ. Tokyo
Vol. 7 (2000), No. 1, Page 99--146.

Sugita, Hiroshi ; Takanobu, Satoshi
A Limit Theorem for Weyl Transformation in Infinite-Dimensional Torus and Central Limit Theorem for Correlated Multiple Wiener Integrals
We show that under many of the probabilities on $\T^{\infty}$, infinite-dimensional torus, a random system $(1/\sqrt{N} \sum_{i=1}^N f(x_i+pÎ±_i))$ converges to a centered Gaussian system whose covariance is determined only by the distribution of $(Î±_i)_{i=1}^{\infty}$ over $\T$. Moreover we show the convergence of a system of symmetric statistics to that of correlated multiple Wiener integrals defined by the Gaussian system. Also we study the central limit theorem for a sequence of the correlated multiple Wiener integrals.