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

The paper analyzes random vectors in an infinite-dimensional Hilbert space under the condition that for every $k$, the joint law of the first $k+1$ vectors is absolutely continuous with respect to a reference measure $Q$ that vanishes on all finite-dimensional subspaces. It leverages Gram determinants and a key recursion $f^k= f^{k-1} h^k{}^2$, where $h^k$ is the distance from the new vector to the span of the previous ones, to prove by induction that the degeneracy set has $Q^{\otimes(k+1)}$-measure zero for each $k$, which implies $\,\det G_k>0$ almost surely and hence almost-sure linear independence (freedom) of the sequence. The results require the finite-dimensional subspaces to be $Q$-negligible, and the authors remark that if the vectors are independent and AC with respect to $Q$, then the joint laws are AC with respect to the product measure; auxiliary appendices justify a Gram-determinant identity and the negligibility of affine finite-dimensional subspaces. Overall, the work provides a rigorous criterion guaranteeing almost-sure independence in infinite dimensions under a specialized Hilbert-space probability framework.

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.

Theorems & Definitions (4)

• Proposition 1.1
• Theorem 1.1
• proof
• Remark 2.1