A Note on Density Estimation for Binary Sequences

A histogram estimate of the Radon-Nikodym derivative of a probability measure with respect to a dominating measure is developed for binary sequences in $\{0,1\}^{\mathbb{N}}$. A necessary and sufficient condition for the consistency of the estimate in the mean-square sense is given. It is noted that the product topology on $\{0,1\}^{\mathbb{N}}$ and the corresponding dominating product measure pose considerable restrictions on the rate of sampling required for the requisite convergence.