Distribution of product of inner products with random normal vector and two fixed vectors

Question

Let $X \sim\mathcal{N}(0, I_n)$ be a random Gaussian vector. Suppose we have two fixed vectors $a,b\in\mathbb{R}^{n}$. We know
\begin{equation}
\langle X, a \rangle \sim \mathcal{N}(0, \|a\|_2^2), \quad \langle X, b \rangle \sim \mathcal{N}(0, \|b\|_2^2),
\end{equation}
and
\begin{equation}
\mathbb{E}\left[\langle X, a \rangle\, \langle X, b \rangle \right] = \langle a, b\rangle.
\end{equation}
I am wondering what else we could say about the distribution of the random variable $\langle X, a \rangle\, \langle X, b \rangle$. Under what conditions are $\langle X, a \rangle$ and $\langle X, b \rangle$ independent? Assuming this is only when $a$ and $b$ are unit orthogonal?

Answer 1

Since these two random variables are both linear combinations of the components of a multivariate Gaussian random vector, they are indepedent if they are uncorrelated. And \begin{align} & \operatorname{cov}(\langle X,a\rangle, \langle X,b\rangle) \\[8pt] = {} & \operatorname{cov}(a^\top X, b^\top X) \\[8pt] = {} & a^\top \operatorname{cov}(X,X) b \\[8pt] = {} & a^\top I_n b \\[8pt] = {} & 0 \text{ if, but only if, } a,b \text{ are orthogonal.} \end{align}