Maxwell's theorem

See Maxwell's theorem (geometry) for the result on triangles.

In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a random vector in $\mathbb {R} ^{n}$ 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, ..., 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.

History

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

Ten years earlier, Herschel also proved the theorem.

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

References

• 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.