Conditional isotropy for nonsymmetric tensor base laws

Determine whether the conditional second-moment relation E[xx^{\mathsf T}\mid R_p]=R_pI_p, or the stated approximate conditional-isotropy condition, holds for the tensor feature vector when the base random variable is not symmetric about zero.

Background

For tensor feature vectors x_I=∏_{i∈I}X_i, symmetry of the base law under X↦−X makes the off-diagonal conditional second moments vanish and yields exact conditional isotropy given the radius. This permits application of the conditional converse theorem.

The paper explicitly leaves unresolved whether an analogous conditional-isotropy property holds for nonsymmetric base laws. Establishing it would extend the conditional spectral characterization to a broader class of tensor models.