Editing Maxwell's theorem

{{Short description|Concept in probability theory}}
{{hatnote|See [[Maxwell's theorem (geometry)]] for the result on triangles.}}
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 <math>\R^n</math> 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 space|vector]]-valued [[random variable]] ''X'' = ( ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> )<sup>''T''</sup> is the same as the distribution of ''GX'' for every ''n''×''n'' [[orthogonal matrix]] ''G'' and the components are [[statistical independence|independent]], then the components ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[normal distribution|normally distributed]] with [[expected value]] 0 and all have the same [[variance]].  This theorem is one of many [[characterization (mathematics)|characterizations]] of the normal distribution.

The only rotationally invariant probability distributions on '''R'''<sup>''n''</sup> that have independent components are [[multivariate normal distribution]]s with [[expected value]] '''0''' and [[variance]] ''σ''<sup>2</sup>''I''<sub>''n''</sub>, (where ''I''<sub>''n''</sub> = the ''n''×''n'' identity matrix), for some positive number ''σ''<sup>2</sup>.

== History ==
[[John Herschel]] proved the theorem in [[1850]].<ref>Herschel, J. F. W. (1850). Quetelet on probabilities. Edinburgh Rev., 92, 1–57.</ref><ref>{{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".</ref> Ten years later, [[James Clerk Maxwell]] proved the theorem in Proposition IV of his 1860 paper.<ref>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.</ref><ref>{{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}}</ref>

== 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, <math>X_1</math> and <math>X_2</math> have the same probability measure: let it be <math>\mu</math>. If <math>\mu</math> is a Dirac delta distribution at zero, then it is in particular a degenerate gaussian distribution. Let us now assume that it is not a Dirac delta distribution at zero.

By the [[Lebesgue's decomposition theorem]], we decompose <math>\mu</math> to a sum of regular measure and an atomic measure: <math>\mu = \mu_r + \mu_s</math>. We need to show that <math>\mu_s = 0</math>; we proceed by contradiction. Suppose <math>\mu_s</math> contains an atomic part, then there exists some <math>x\in \R</math> such that <math>\mu_s(\{x\}) > 0</math>. By independence of <math>X_1, X_2</math>, the conditional variable <math>X_2 | \{X_1 = x\}</math> is distributed the same way as <math>X_2</math>. Suppose <math>x=0</math>, then since we assumed <math>\mu</math> is not concentrated at zero, <math>Pr(X_2 \neq 0) > 0</math>, and so the double ray <math>\{(x_1, x_2): x_1 = 0, x_2 \neq 0\}</math> has nonzero probability. Now, by rotational symmetry of <math>\mu \times \mu</math>, any rotation of the double ray also has the same nonzero probability, and since any two rotations are disjoint, their union has infinite probability; thus arriving at a contradiction.

Let <math>\mu </math> have probability density function <math>\rho</math>; the problem reduces to solving the functional equation

<math display="block">\rho(x)\rho(y) = \rho(x \cos \theta + y \sin\theta)\rho(x \sin \theta - y \cos\theta).</math>

==References==
{{reflist}}

==Sources==
* {{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}}

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

{{DEFAULTSORT:Maxwell's Theorem}}
[[Category:Theorems in probability theory]]
[[Category:James Clerk Maxwell]]