Strong law of large numbers

The strong law of large numbers is a mathematical proposition from probability theory that makes statements about when a sequence of normalized random variables converges against a constant, usually the expected value of the random variable. The strong law of large numbers is counted with the weak law of large numbers to the laws of large numbers and belongs to the classic limit theorems of stochastics . The difference between the “strong” and the “weak” version is the type of convergence under consideration: The strong law of large numbers makes a statement about the P-almost certain convergence of the random variables, the weak law of large numbers, however, about the stochastic convergence of the random variables.

A distinction is often made between several versions of the strong law of large numbers, which differ in the generality of their formulation or the strength of their presuppositions. For example, there is Borel's strong law of large numbers (after Émile Borel ), Kolmogorov's first and second strong law of large numbers (after Andrei Nikolajewitsch Kolmogorow ) or Etemadi's strong law of large numbers (after Nasrollah Etemadi ).

formulation

If a sequence of random variables is given, it is said that this sequence satisfies the strong law of large numbers if the mean value of the scaled random variables ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

${\displaystyle {\overline {X}}_{n}:={\frac {1}{n}}\sum _{i=1}^{n}\left(X_{i}-\operatorname {E} (X_{i})\right)}$

almost certainly converged to 0 . It means that

${\displaystyle P(\lim _{n\to \infty }{\overline {X}}_{n}=0)=1}$

is.

Interpretation and difference to the weak law of large numbers

→ Main article : Laws of large numbers: interpretation of the statements

The weak law of large numbers always follows from the strong law of large numbers .

validity

Below are various conditions under which the strong law of large numbers applies. The weakest and most specific statement is at the top, the strongest and most general at the bottom.

Borel's strong law of large numbers

If there is a sequence of independent random variables distributed to the parameter Bernoulli , then the sequence satisfies the strong law of large numbers, that is, the mean value of the random variable almost certainly converges to the parameter . ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle p\in (0,1)}$ ${\displaystyle p}$

This statement was proven by Émile Borel in 1909 and corresponds to the formulation of Bernoulli's Law of Large Numbers as a strong law of large numbers.

Cantelli's theorem

→ Main article : Cantelli's theorem

Cantelli's theorem provides the validity of the strong law of large numbers under requirements of the fourth moments and the centered fourth moments. It was proved by Francesco Paolo Cantelli in 1917 and is considered to be the first result that provides the validity of the strong law of large numbers for sequences of random variables of any distribution.

Kolmogorov's First Law of Large Numbers

Is an independent sequence of random variables with finite variance given and holds ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {\operatorname {Var} (X_{n})}{n^{2}}}<\infty }$,

so suffices the strong law of large numbers. The above condition is also called Kolmogorow's condition . and is used to estimate using the Kolmogorow inequality . ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

This statement was proven in 1930 by Andrei Nikolayevich Kolmogorov .

Second Law of Large Numbers by Kolmogorov

If the sequence of random variables is independently identically distributed , and then the sequence satisfies the strong law of large numbers. ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \operatorname {E} (|X_{n}|)<\infty }$

The statement was proven in 1933 by Andrei Nikolayevich Kolmogorov.

Strong law of the great number of Etemadi

If the sequence of random variables is identically distributed and independently in pairs , then it satisfies the strong law of large numbers. ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \operatorname {E} (|X_{n}|)<\infty }$

This statement is a real improvement over Kolmogorov's second law of large numbers, since independence always implies independence in pairs, but the conclusion is generally not valid. This statement was proven by Nasrollah Etemadi in 1981 .

Alternative formulations

More general wording

Somewhat more generally, we say that the sequence of random variables satisfies the strong law of large numbers if there are real sequences with and , so that for the partial sum ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \lim _{n\to \infty }b_{n}=\infty }$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$

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

the convergence

${\displaystyle {\frac {S_{n}}{b_{n}}}-a_{n}\rightarrow 0}$

almost certainly applies.

More specific formulation

Some authors consider the almost certain convergence of the averaged partial sums against . However, this formulation assumes that all random variables have the same expected value. ${\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$ ${\displaystyle \operatorname {E} (X_{1})}$

Generalizations

Ergodic sentences

A possible generalization of the strong law of large numbers is the individual ergodic set and the Lp-ergodic set . These results of the ergodic theory can be applied to stationary stochastic processes . Thus, with these sentences a stochastic dependence of the observed sequence on random variables is possible. ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

Vector-valued illustrations

The law of large numbers can also be formulated for vector-valued images. Mourier's theorem provides some criteria for this .

Web links

• Yu.V. Prokhorov: Strong law of large numbers . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Strong law of large numbers . In: MathWorld (English).

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 .
• Hans-Otto Georgii : Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Christian Hesse : Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .

Individual evidence

1. ↑ Hesse: Applied probability theory. 2003, p. 249.
2. ^ AV Prokhorov: Borel strong law of large numbers . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
3. ↑ Yu.V. Prokhorov: Strong law of large numbers . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
4. ↑ Kusolitsch: Measure and probability theory. 2014, p. 251.
5. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 157.
6. ^ Schmidt: Measure and Probability. 2011, p. 347.
7. ↑ Klenke: Probability Theory. 2013, p. 114.
8. ^ Nasrollah Etemadi: An elementary proof of the strong law of large numbers. In: Journal of Probability Theory and Allied Areas. (Online edition: Probability Theory and Related Fields. Continuation of Zeitschrift fur Probabilstheorie. ). Vol. 55, No. 1, 1981, pp. 119-122, doi : 10.1007 / BF01013465 .
9. ↑ Hesse: Applied probability theory. 2003, p. 249.