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The convolution theorem does not extend to cylindrical measures on separable Hilbert spaces

By W. Schachermayer, H. Strasser and W. Schachermayer Und H. Strasser


In this paper we give an example which shows that the convolution theorem (Boll, [1], Hajek, [2]) cannot be extended to innite-dimensional shift experiments. This answers a question posed by van der Vaart, [8], and which has been considered also by LeCam, [4]. 1 Introduction Let H be an nite dimensional vector space. Assume that P is a probability measure on the Borel-#-eld of H, and that A : H # R is a continuous linear function. Let T : H # R be a measurable function. The image of P under T is denoted by T (P ). The symbol # h denotes the point measure at h # H. The measurable function T : H # R is called an equivariant estimator of the function A, if T (P # # h ) = T (P ) # # Ah for h # H. The assertion of the convolution theorem states that under these conditions there exists a probability measure R such that T (P ) = A(P ) # R. In this form the convolution theorem has been proved by Boll, [1]. For the history of research on the convolution theorem we refer t..

Year: 2000
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