Optimal properties of the canonical tight probabilistic frame

A probabilistic frame is a Borel probability measure with finite second moment whose support spans $\mathbb{R}^d$. A Parseval probabilistic frame is one for which the associated matrix of the second moments is the identity matrix in $\mathbb{R}^d$. Each probabilistic frame is canonically associated to a Parseval probabilistic frame. In this paper, we show that this canonical Parseval probabilistic frame is the closest Parseval probabilistic frame to a given probabilistic frame in the $2-$Wasserstein distance. Our proof is based on two main ingredients. On the one hand, we show that a probabilistic frame can be approximated in the $2-$Wasserstein metric with (compactly supported) finite frames whose bounds can be controlled. On the other hand, we establish some fine continuity properties of the function that maps a probabilistic frame to its canonical Parseval probabilistic frame. Our results generalize similar ones for finite frames and their associated Parseval frames.