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Approximation of \(L_2\)-processes by Gaussian processes. (English) Zbl 0914.28013

Let \((X,{\mathcal F},\mu)\) be a nonatomic probability space, \(T\) an ergodic transformation on \(X\), \(f\in L_2(\mu)\) and \(K\geq 1\) an integer. It is well known that the \(L_2\)-process \((T,f)= (T^if)\) is \(L_2\)-equivalent to a unique Gauss process \((V,h)\). The authors consider two notions of closeness of \(L_2\)-processes:
(i) \(L_2\)-closeness within \((K,\varepsilon)\) – if the inner products \((T^if, T^jf)\) and \((T^ig, T^jg)\) are \(\varepsilon\)-close for \(1\leq i, j\leq K\),
(ii) weak-closeness with \((K,\varepsilon)\) – if their \(K\)-dimensional distribution measures are \(\varepsilon\)-close.
The main theorem in this paper shows that if \(T\) is aperiodic, and \((T,f)\) is an \(L_2\) process with \((V,h)\) the corresponding unique Gauss process which is \(L_2\)-equivalent to \((T,f)\), then there exists a \(g\in L_2(\mu)\) such that \((V,h)\) and \((T,g)\) are both weakly and \(L_2\) close within \((K,\varepsilon)\).
As a corollary of the main theorem, it is shown that if \(A_1,\dots, A_K\) are \(K\)-operators which are strong limits of linear combinations of iterates of \(T\), and if \(q_i\), \(1\leq i\leq K\), are the Gauss sequences \(L_2\)-equivalent to \(A_if\), then there exists a \(g\in L_2(\mu)\) such that \(q_i\) and \(A_ig\) are both weakly and \(L_2\) close within \((K,\varepsilon)\). Further, the authors apply this corollary to give a simpler and more elegant proof of a result in [M. A. Akcoglu, D. M. Ha and R. L. Jones, Can. J. Math. 49, No. 1, 3-23 (1997; Zbl 0870.28007)] which is a refinement of Bourgain’s entropy theorem from [J. Bourgain, Isr. J. Math. 63, No. 1, 79-97 (1988; Zbl 0677.60042)].

MSC:

28D05 Measure-preserving transformations
60G15 Gaussian processes
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