Maxwell's theorem

In probability theory, Maxwell's theorem (known also as Herschel-Maxwell's theorem and Herschel-Maxwell's derivation) states that if the probability distribution of a random vector in $$\R^n$$ is unchanged by rotations, and if the components are independent, then the components are identically distributed and normally distributed.

Equivalent statements
If the probability distribution of a vector-valued random variable X = ( X1, ..., Xn )T is the same as the distribution of GX for every n×n orthogonal matrix G and the components are independent, then the components X1, ..., Xn are normally distributed with expected value 0 and all have the same variance. This theorem is one of many characterizations of the normal distribution.

The only rotationally invariant probability distributions on Rn that have independent components are multivariate normal distributions with expected value 0 and variance σ2In, (where In = the n×n identity matrix), for some positive number σ2.

History
James Clerk Maxwell proved the theorem in Proposition IV of his 1860 paper.

Ten years earlier, John Herschel also proved the theorem.

The logical and historical details of the theorem may be found in.

Proof
We only need to prove the theorem for the 2-dimensional case, since we can then generalize it to n-dimensions by applying the theorem sequentially to each pair of coordinates.

Since rotating by 90 degrees preserves the joint distribution, both $$X_1, X_2$$ has the same probability measure. Let it be $$\mu$$. If $$\mu$$ is a Dirac delta distribution at zero, then it's a gaussian distribution, just degenerate. Now assume that it is not.

By Lebesgue's decomposition theorem, we decompose it to a sum of regular measure and an atomic measure: $$\mu = \mu_r + \mu_s$$. We need to show that $$\mu_s = 0$$, with a proof by contradiction.

Suppose $$\mu_s$$ contains an atomic part, then there exists some $$x\in \R$$ such that $$\mu_s(\{x\}) > 0$$. By independence of $$X_1, X_2$$, the conditional variable $$X_2 | \{X_1 = x\}$$ is distributed the same way as $$X_2$$. Suppose $$x=0$$, then since we assumed $$\mu$$ is not concentrated at zero, $$Pr(X_2 \neq 0) > 0$$, and so the double ray $$\{(x_1, x_2): x_1 = 0, x_2 \neq 0\}$$ has nonzero probability. Now by rotational symmetry of $$\mu \times \mu$$, any rotation of the double ray also has the same nonzero probability, and since any two rotations are disjoint, their union has infinite probability, contradiction.

(As far as we can find, there is no literature about the case where $$\mu_s$$ is singularly continuous, so we will let that case go.)

So now let $$\mu $$ have probability density function $$\rho$$, and the problem reduces to solving the functional equation

$$\rho(x)\rho(y) = \rho(x \cos \theta + y \sin\theta)\rho(x \sin \theta - y \cos\theta)$$