Remark on the finite-dimensional character of certain results of functional statistics

TL;DR

This paper demonstrates that certain assumptions in functional statistics imply the underlying space is finite-dimensional, supported by an example of an L2 process that violates these assumptions.

Contribution

It reveals that common small balls probability assumptions in functional statistics enforce finite-dimensionality of the space, clarifying limitations of these assumptions.

Findings

Small balls probability assumptions imply finite-dimensionality.

An example of an L2 process violating these assumptions is provided.

Highlights limitations of certain assumptions in functional statistics.

Abstract

This note shows that some assumption on small balls probability, frequently used in the domain of functional statistics, implies that the considered functional space is of finite dimension. To complete this result an example of L2 process is given that does not fulfill this assumption.