below. However, even though the proof of Theorem \[abstract\] is “conceptually" simple, the technical details make it rather long. This would be exactly the same for the strong mixing case (i.e. part (2) of Theorem \[WS\]), but in the weak mixing case it is possible to give a technically much simpler and hence much shorter proof. For the sake of readability, it seems desirable to present this proof separately. But since there is no point in repeating identical arguments, we shall follow the abstract approach as long as this does not appear to be artificial.

The paper is organized as follows. In section 2, we present our abstract results. In section 3, we review some basic facts concerning Gaussian measures and we outline the strategy for proving the abstract results and hence Theorem \[WS\]. Apart from some details in the presentation and the level of generality, this follows the scheme described in [@BG3], [@BG2] and [@BM]. In section 4, we prove part (1) of Theorem \[WS\] (the sufficient condition for weak mixing). The abstract results are proved in sections \[proofabstract1\] and \[proofabstract2\]. Section \[final\] contains some additional examples and miscellaneous remarks. In particular, we briefly discuss the “continous" analogues of 