Maxwell's theorem

If the probability distribution of a vector-valued random variable X = ( X₁, ..., X_n )^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, ..., X_n 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 σ²I_n, (where I_n = the n×n identity matrix), for some positive number σ².

James Clerk Maxwell proved the theorem in Proposition IV of his 1860 paper.[1]

Ten years earlier, John Herschel also proved the theorem.[2]

The logical and historical details of the theorem may be found in.[3]

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 ${\displaystyle X_{1},X_{2}}$ has the same probability measure. Let it be ${\displaystyle \mu }$. If ${\displaystyle \mu }$ 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: ${\displaystyle \mu =\mu _{r}+\mu _{s}}$. We need to show that ${\displaystyle \mu _{s}=0}$, with a proof by contradiction.

Suppose ${\displaystyle \mu _{s}}$ contains an atomic part, then there exists some ${\displaystyle x\in \mathbb {R} }$ such that ${\displaystyle \mu _{s}(\{x\})>0}$. By independence of ${\displaystyle X_{1},X_{2}}$, the conditional variable ${\displaystyle X_{2}|\{X_{1}=x\}}$ is distributed the same way as ${\displaystyle X_{2}}$. Suppose ${\displaystyle x=0}$, then since we assumed ${\displaystyle \mu }$ is not concentrated at zero, ${\displaystyle Pr(X_{2}\neq 0)>0}$, and so the double ray ${\displaystyle \{(x_{1},x_{2}):x_{1}=0,x_{2}\neq 0\}}$ has nonzero probability. Now by rotational symmetry of ${\displaystyle \mu \times \mu }$, 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 ${\displaystyle \mu _{s}}$ is singularly continuous, so we will let that case go.)

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

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

• Feller, William (1966). An Introduction to Probability Theory and its Applications. Vol. II (1st ed.). Wiley. p. 187.
• Maxwell, James Clerk (1860). "Illustrations of the dynamical theory of gases". Philosophical Magazine. 4th Series. 19: 390–393.

• Maxwell's theorem in a video by 3blue1brown