Misplaced Pages

Maxwell's theorem

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Concept in probability theory 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 R n {\displaystyle \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, ..., Xn ) is the same as the distribution of GX for every n×n orthogonal matrix G and the components are independent, then the components X1, ..., Xn 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 R that have independent components are multivariate normal distributions with expected value 0 and variance σIn, (where In = the n×n identity matrix), for some positive number σ.

History

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

Ten years earlier, John Herschel also proved the theorem.

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

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

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

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

ρ ( x ) ρ ( y ) = ρ ( x cos ⁡ θ + y sin ⁡ θ ) ρ ( x sin ⁡ θ − y cos ⁡ θ ) {\displaystyle \rho (x)\rho (y)=\rho (x\cos \theta +y\sin \theta )\rho (x\sin \theta -y\cos \theta )}

References

  1. See:
  2. Herschel, J. F. W. (1850). Quetelet on probabilities. Edinburgh Rev., 92, 1–57.
  3. Gyenis, Balázs (February 2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency toward equilibrium". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 57: 53–65. arXiv:1702.01411. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001. ISSN 1355-2198. S2CID 38272381.

Sources

External links

Categories: