Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures

TL;DR

This paper proves that infinite-dimensional Hilbert space-valued random vectors are almost surely linearly independent if their finite-dimensional projections are absolutely continuous with respect to a special class of probability measures, with no restrictions on their span.

Contribution

It establishes almost sure linear independence of Hilbert space-valued vectors under broad conditions, extending previous results to a special class of probability measures.

Findings

Vectors are almost surely linearly independent under the given conditions.

Finite-dimensional subspaces are negligible with respect to the probability measure.

No constraints on the random dimension of the span are needed.

Abstract

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a probability measure. It demonstrates that no constraints on the random dimension of their span are necessary, provided that all finite-dimensional vector subspaces are considered negligible with respect to the Hilbert space probability measure.