Optimal unbiased estimators in additive models with bounded errors are deterministic

In an additive model X = v + epsilon, v is an element of theta subset of R(k), let the errors epsilon have a compactly supported but otherwise arbitrary known joint distribution. Let g be a uniformly minimum variance unbiased estimator for its own expectation gamma(v). We show that under mild regularity conditions, g is deterministic: for every v is an element of theta, g(X) = gamma(v) almost surely. Our proof uses a lemma on entire quotients of Fourier transforms which might be of independent interest.