Cantelli's theorem

The set of Cantelli is a theorem of probability theory , one of the branches of mathematics . It goes back to the Italian mathematician Francesco Paolo Cantelli and formulates a sufficient condition for the existence of the strong law of large numbers for certain sequences of real random variables . The Cantellian theorem is considered to be one of the first results of its kind.

Formulation of the sentence

The Cantellian sentence can be specified as follows:

A probability space and a sequence of random variables are given ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$

${\displaystyle X_{n}\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} \;(n\in \mathbb {N} )}$

on this probability space.

Let the consequence be stochastically independent and with finite fourth moments :

${\displaystyle \operatorname {E} {\bigl (}{X_{n}}^{4}{\bigr )}<{\infty }\;(n\in \mathbb {N} )}$  .

In addition, let the central fourth moments be uniformly bounded upwards :

${\displaystyle \sup _{n\in \mathbb {N} }\;{\operatorname {E} {\bigl (}{(X_{n}-\operatorname {E} (X_{n}))}^{4}{\bigr )}}<{\infty }}$  .

Then it is sufficient to result - almost certainly the convergence ${\displaystyle \operatorname {P} }$

${\displaystyle \lim _{n\to \infty }{{\frac {1}{n}}\sum _{j=1}^{n}{{\bigl (}X_{j}-{\operatorname {E} (X_{j})}{\bigr )}}}=0}$

and with it the strong law of large numbers.

Proof of the Širjaev theorem

You bet for ${\displaystyle j\in \mathbb {N} }$

${\displaystyle Y_{j}=X_{j}-{\operatorname {E} (X_{j})}}$

and continue for ${\displaystyle n\in \mathbb {N} }$

${\displaystyle S_{n}=\sum _{j=1}^{n}{Y_{j}}}$

such as

${\displaystyle C=\sup _{n\in \mathbb {N} }\;{\operatorname {E} {\bigl (}{Y_{n}}^{4}{\bigr )}}}$

Then is for ${\displaystyle n\in \mathbb {N} }$

(0) ${\displaystyle \quad \operatorname {E} {(Y_{n})}=0=\operatorname {E} {(S_{n})}}$

and consequently it has to be shown that

(1) ${\displaystyle \quad \lim _{n\to \infty }{\frac {S_{n}}{n}}=0\;(\operatorname {P} {\mathrm {\text{-fast sicher}} })}$

applies.

If one now takes into account the conclusion mentioned in the last section of the article on the Borel-Cantelli lemma and the Chebyshev-Markov inequality , one sees that the convergence of the series is sufficient

(2) ${\displaystyle \quad \sum _{n=1}^{\infty }{\frac {\operatorname {E} {({S_{n}}^{4}})}{n^{4}}}}$

to prove.

To do this, the terms of the series (2) are evaluated using the polynomial theorem .

It is namely:

(3)  . ${\displaystyle \quad {S_{n}}^{4}={(Y_{1}+\ldots +Y_{n})}^{4}=\sum _{k_{1}+\ldots +k_{n}=4}{{\frac {4!}{k_{1}!\cdot \ldots \cdot k_{n}!}}\cdot {Y_{1}}^{k_{1}}\cdot {Y_{2}}^{k_{2}}\cdots {Y_{n}}^{k_{n}}}}$

When forming the expected values for (3), only those summands are important for which   the exponents   or     occur exclusively in the associated   ones . ${\displaystyle Y_{j}}$   ${\displaystyle 2}$ ${\displaystyle 4}$

Because in all other cases there is at least one     with exponent     and because of the linearity of the expected value, the independence requirement and because of   (0)   in the expected value of (3) only the summands with even exponents make a positive contribution. ${\displaystyle Y_{j}}$ ${\displaystyle 1}$

So you have

(4)  . ${\displaystyle \quad \operatorname {E} {({S_{n}}^{4})}=\sum _{j=1}^{n}{\operatorname {E} {({Y_{j}}^{4})}}\;+\;\sum _{i,j=1,\ldots ,n \atop \;\ i<j}{{\frac {4!}{2!\cdot 2!}}\cdot \operatorname {E} {({Y_{i}}^{2})}\cdot \operatorname {E} {({Y_{j}}^{2})}}}$

With (4) and applying the assumption and Lyapunov's inequality, the following chain of inequalities results :

(5) ${\displaystyle \quad {\begin{aligned}\operatorname {E} {({S_{n}}^{4})}&\leq n\cdot C\;+\;6\cdot \sum _{i,j=1,\ldots ,n \atop \;\ i<j}{{\operatorname {E} {({Y_{i}}^{4})}}^{\frac {1}{2}}\cdot {\operatorname {E} {({Y_{j}}^{4})}}^{\frac {1}{2}}}\leq n\cdot C\;+\;6\cdot \sum _{i,j=1,\ldots ,n \atop \;\ i<j}{{C}^{\frac {1}{2}}\cdot {C}^{\frac {1}{2}}}\\&=n\cdot C\;+\;6\;\cdot {\frac {n\cdot {(n-1)}}{2}}\cdot C=(3\cdot n^{2}-2\cdot n)\cdot C<3\cdot n^{2}\cdot C.\end{aligned}}}$

The chain of inequalities (5) draws in consideration of the convergence of the zeta-series part, the chain of inequalities

(6) ${\displaystyle \quad \sum _{n=1}^{\infty }{\frac {\operatorname {E} {({S_{n}}^{4}})}{n^{4}}}\leq 3\cdot C\cdot \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}=3\cdot C\cdot {\frac {\pi ^{2}}{6}}<{\infty }}$

after itself and thus also (2) .

${\displaystyle \Box }$

literature

• AN Širjaev : Probability (=  university books for mathematics . Volume 91 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1988, ISBN 3-326-00195-9 ( MR0967761 ).
• Eugenio Regazzini: Probability and statistics in Italy during the First World War . I: Cantelli and the laws of large numbers. In: Electronic Journ @ l for History of Probability and Statistics . tape 1 , 2005, p. 1-12 ( jehps.net [PDF] MR2208347 ).

References and comments

1. ↑ TO Širjaev: Probability. 1988, pp. 379-380
2. ↑ For a real random variable , its expected value is designated. ${\displaystyle \xi }$ ${\displaystyle \operatorname {E} {\bigl (}\xi {\bigr )}}$