The Convolution Theorem of Hajek and Le Cam - Revisited

TL;DR

This paper extends the convolution theorem to non-Gaussian and infinite-dimensional limit experiments, providing new theoretical insights and applications in statistical estimation for complex stochastic models.

Contribution

It introduces convolution theorems for regular estimators in non-Gaussian and infinite-dimensional settings, with an elementary proof method for comparing limit experiments.

Findings

Convolution theorems established for Gaussian shift experiments of infinite dimension.

Applications to Brownian motion signals, Levy processes, and density endpoint estimation.

Provides a new elementary approach for comparing limit experiments.

Abstract

The present paper establishes convolution theorems for regular estimators when the limit experiment is non-Gaussian or of infnite dimension with sparse parameter space. Applications are given for Gaussian shift experiments of infnite dimension, the Brownian motion signal plus noise model, Levy processes which are observed at discrete times and estimators of the endpoints of densities with jumps. The method of proof is also of interest for the classical convolution theorem of Hajek and Le Cam. As technical tool we present an elementary approach for the comparison of limit experiments on standard Borel spaces.