Let me start by setting the scene and stating the theorem. I just want some pointers on how to follow the proof.
Let $y$ be an observable, $n$-dimensional random vector and $f$ be a known, Borel measurable, vector function such that $$ f: \mathbb{R}^k \times \mathbb{R}^m \rightarrow \mathbb{R}^n$$
where $\theta$ is a $k$-dimensional parameter vector which may be unknown and $v$ is some unobservable $m$-dimensional random vector. We consider a potentially nonlinear model $$ y= f(\theta, v) $$ and define $\sigma^2 > 0$ as a scalar and $\Sigma$ as any positive definite $m \times m$ matrix.
Theorem: With respect to the above model, any statistics which is invariant to the scale of $v$ has the same distribution when $v \sim N(0, \sigma^2 \Sigma)$ as it does for $v$ taking any other elliptically symmetrical distribution $E_o(m, \Sigma)$.
Proof: Denote the statistic by $g(y)$. Define the function, $\bar{g}: \mathbb{R}^m \rightarrow \mathbb{R}$ par $\bar{g}(v) = g\left( f(\theta,v) \right)$ and let $G$ be the group of transformations of the form $v \rightarrow \lambda v$ where $\lambda$ is any positive scalar.
Consider the statistic $\eta(v) = v/ \left( v' \Sigma^{-1} v \right)^{1/2}$. Clearly $\eta(v)$ is invariant to transformations belonging to $G$. If $v_{(1)}$ and $v_{(2)}$ are two $m$-dimensional vectors such that $\eta (v_{(1)}) = \eta (v_{(2)})$, then $v_{(1)}$ and $v_{(2)}$ are related by a transformation belonging to $G$. Therefore, $\eta(v)$ is maximal invariant. Because $\bar{g}$ is also invariant to transformations belonging to $G$, it can be written as a function of $\eta(v)$, i.e., \begin{align} g(y) = \bar{g}(v) = g^*\left( v/ \left( v' \Sigma^{-1} v \right)^{1/2} \right), \end{align} say. When $v$ takes an $E_o(m,\Sigma)$ distribution, $\Sigma^{-1/2}v$ is $E_o(m,I_m)$ and by property 10, $\Sigma^{-1/2}v/ \left( v' \Sigma^{-1} v \right)^{1/2}$ is uniformly distributed on the surface of the $n$-dimensional unit sphere. Thus $v/ \left( v' \Sigma^{-1} v \right)^{1/2}$ and hence $g(y)$ have the same distributions for any $E_o(m,\Sigma)$ distribution followed by $v$, including $N(0,\sigma^2 \Sigma)$. $\blacksquare$
Property 10: If $x$ is an $E_o(n,I_n)$ random vector then $x/(x'x)^{1/2}$ is uniformly distribution on the $n$-dimensional unit hypersphere $C_n = \{ x|x \in \mathbb{R}^r, x'x = 1\}$.
The theorem here is from Maxwell L. King's 1979 PhD thesis. Now, let me try to understand the proof. The first step seems to involve showing that any arbitrary statistic is going to be some function of $\eta(v)$. The second step says that regardless of the distribution of $v$, provided it is in this class of distribution, the distribution of $\eta(v)$ is going to be exactly the same. Finally, because $N(0, \sigma^2 \Sigma)$ is in that class, the proof is completed.
I am not too familiar with some of the concepts involved, so I'd like to know if I got at least the outline right and, then, maybe some explanations about each parts.
