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.

John Herschel proved the theorem in 1850.Herschel, J. F. W. (1850). [https://www.google.com/books/edition/Essays_from_the_Edinburgh_and_Quarterly/S48UtJaeE-8C?hl=en&gbpv=1&dq=%22quetelet%20on%20probabilities%22&pg=PA365&printsec=frontcover Review of Quetelet on probabilities]. Edinburgh Rev., 92, 1–57.{{harvtxt|Bryc|1995|p=1}} quotes Herschel and "state[s] the Herschel-Maxwell theorem in modern notation but without proof". Bryc cites M. S. Bartlett (1934) "for one of the early proofs" and lists several variants of the theorem that are proven in his book. Ten years later, James Clerk Maxwell proved the theorem in Proposition IV of his 1860 paper.See:

• Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&pg=PA19 "Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,"] Philosophical Magazine, 4th series, 19 : 19–32.
• Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&pg=PA21 "Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another,"] Philosophical Magazine, 4th series, 20 : 21–37.{{Cite journal |last=Gyenis |first=Balázs |date=February 2017 |title=Maxwell and the normal distribution: A colored story of probability, independence, and tendency toward equilibrium |url=http://dx.doi.org/10.1016/j.shpsb.2017.01.001 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |volume=57 |pages=53–65 |doi=10.1016/j.shpsb.2017.01.001 |arxiv=1702.01411 |bibcode=2017SHPMP..57...53G |s2cid=38272381 |issn=1355-2198}}

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

Since rotating by 90 degrees preserves the joint distribution, $X\_1$ and $X\_2$ have the same probability measure: let it be $\backslash mu$. If $\backslash mu$ is a Dirac delta distribution at zero, then it is, in particular, a degenerate gaussian distribution. Let us now assume that $\backslash mu$ is not a Dirac delta distribution at zero.

By Lebesgue's decomposition theorem, we can decompose $\backslash mu$ into a sum of a regular measure and an atomic measure: $\backslash mu\; =\; \backslash mu\_r\; +\; \backslash mu\_s$. We need to show that $\backslash mu\_s\; =\; 0$; we proceed by contradiction. Suppose $\backslash mu\_s$ contains an atomic part, then there exists some $x\backslash in\; \backslash R$ such that $\backslash mu\_s(\backslash \{x\backslash \})\; >\; 0$. By independence of $X\_1,\; X\_2$, the conditional variable $X\_2\; |\; \backslash \{X\_1\; =\; x\backslash \}$ is distributed the same way as $X\_2$. Suppose $x=0$, then since we assumed $\backslash mu$ is not concentrated at zero, $Pr(X\_2\; \backslash neq\; 0)\; >\; 0$, and so the double ray $\backslash \{(x\_1,\; x\_2):\; x\_1\; =\; 0,\; x\_2\; \backslash neq\; 0\backslash \}$ has nonzero probability. Now, by the rotational symmetry of $\backslash mu\; \backslash times\; \backslash mu$, any rotation of the double ray also has the same nonzero probability, and since any two rotations are disjoint, their union has infinite probability, which is a contradiction.{{clarify|reason=State the obvious by explicitly describing the contradiction.|date=June 2025}}

Let $\backslash mu$ have probability density function $\backslash rho$; the problem reduces to solving the functional equation

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

{{reflist}}

• {{cite book |title=The Normal Distribution: Characterizations with Applications|last=Bryc|first=Wlodzimierz|publisher=Springer-Verlag|year=1995|isbn=978-0-387-97990-8|url=https://www.google.com/books/edition/The_Normal_Distribution/tyXjBwAAQBAJ?hl=en&gbpv=0}}
• {{cite book|last=Feller|first=William| authorlink=William Feller| date=1966 |title= An Introduction to Probability Theory and its Applications| volume=II| edition=1st| publisher=Wiley| page=187}}
• {{cite journal|last=Maxwell|first=James Clerk|authorlink=James Clerk Maxwell|date=1860 |title=Illustrations of the dynamical theory of gases| journal=Philosophical Magazine |series=4th Series| volume=19| pages=390–393}}

• [https://www.youtube.com/watch?v=cy8r7WSuT1I Maxwell's theorem in a video by 3blue1brown]

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Category:Theorems in probability theory

Category:James Clerk Maxwell