## 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

### Citations:

Zbl 0870.28007; Zbl 0677.60042
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