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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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