Multi-dimensional central limit theorem

The multi-dimensional central limit theorem , also called the central limit theorem ${\displaystyle \mathbb {R} ^{d}}$ or multivariate central limit theorem , is a mathematical proposition from the theory of probability . It belongs to the central limit theorems , generalizes the central limit theorem of Lindeberg-Lévy to higher dimensions and deals with the convergence in distribution of rescaled sums of random vectors against the multidimensional normal distribution .

statement

Given is a sequence of independently identically distributed random vectors in with the zero vector as the expected value vector and a positively definite covariance matrix . ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \mathbf {0} }$ ${\displaystyle C\in \mathbb {R} ^{d\times d}}$

Then the sequence of the rescaled sums converges

${\displaystyle S_{n}:={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}X_{i}}$

in distribution against a random vector which is -dimensionally normally distributed with an expected value vector and a covariance matrix . ${\displaystyle X}$ ${\displaystyle d}$ ${\displaystyle \mathbf {0} }$ ${\displaystyle C}$

Evidence sketch

One possibility of the proof reduces the -dimensional case to the one-dimensional case. For any was ${\displaystyle d}$ ${\displaystyle t\in \mathbb {R} ^{d}}$

${\displaystyle X_{n}^{t}:=\langle t;X_{n}\rangle }$.

The standard scalar product denotes . Then ${\displaystyle \langle \cdot ;\cdot \rangle }$

${\displaystyle \operatorname {E} (X_{n}^{t})=0}$and . ${\displaystyle \operatorname {Var} (X_{n}^{t})=\langle t;Ct\rangle }$

So for all of them, according to Lindeberg-Lévy's central limit theorem, the sequence converges to a real random variable that is normally distributed with an expectation value of 0 and variance . According to Cramér-Wold's theorem , this is equivalent to convergence in distribution of the sequence of random vectors. ${\displaystyle t\in \mathbb {R} ^{d}}$ ${\displaystyle X_{n}^{t}}$ ${\displaystyle X^{t}}$ ${\displaystyle \langle t;Ct\rangle }$

The fact that the sequence of vectors converges to the multidimensional normal distribution follows from the fact that a random vector is multidimensional normally distributed if and only if they are one-dimensionally normally distributed for all (with a suitable expectation and variance). ${\displaystyle X}$ ${\displaystyle X^{t}=\langle t;X\rangle }$ ${\displaystyle t\in \mathbb {R} ^{d}}$

Web links

• Yu.V. Prokhorov: Central limit theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .

Individual evidence

1. ↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 27-28 , doi : 10.1007 / 978-3-642-17261-8 .