# If a probability measure is a mixture of products of its marginals, does it have finite moments?

Let $$\mu$$ be a Borel probability measure on $$\mathbb{R}^n$$. For a linear subspace $$E\subset \mathbb{R}^n$$, let $$\mu_E$$ denote the marginal of $$\mu$$ on $$E$$. The usual orthogonal complement of $$E$$ is denoted $$E^{\perp}$$.

Suppose $$(E_i)_{i=1}^k$$ are linear subspaces of $$\mathbb{R}^n$$ such that

1. $$\cap_{i=1}^k(E_i \cup E_i^{\perp})=\{0\}$$; and
2. $$\mu$$ is equal to the mixture $$\mu = \sum_{i=1}^k p_i \mu_{E_i}\otimes \mu_{E_i^{\perp}}$$, where $$p_i >0$$ for each $$i = 1, \dots, k$$.

Does $$\mu$$ have finite logarithmic moment: $$\int_{\mathbb{R}^n} \log(1+ \|x\|)d\mu(x) <\infty$$?

This is a somewhat simplified version of a question I posed in this manuscript. If the answer to the question is affirmative, then it will imply $$\mu$$ is Gaussian. Some further remarks are given in the appendix of that manuscript. It seems like a nice problem in its own right, so I thought I'd post here.