Maxwell's theorem

In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that 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, all have the same variance, and all are independent. This theorem is one of many characterizations of the normal distribution.

Since a multiplication by an orthogonal matrix is a rotation, the theorem says that if the probability distribution of a random vector is unchanged by rotations and if the components are independent, then the components are identically distributed and normally distributed. In other words, 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.

References

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