Lemma from Frank

The lemma Frank is a mathematical theorem in the field of probability theory , which the mathematician Ove Frank back. It formulates an elementary stochastic inequality for certain finite families of integrable real random variables and thus proves to be a useful tool for the proof of some results in the context of the law of large numbers . With the help of Frank's lemma, the Kolmogoroff inequality and the Chebyshev inequality can be derived.

Formulation of the lemma

Following the presentation by Heinz Bauer , the lemma can be stated as follows:

A probability space and a finite number of integrable random variables are given ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$ ${\displaystyle \operatorname {P} }$ ${\displaystyle Z_{0},Z_{1},\ldots ,Z_{n}\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} \;(n\in \mathbb {N} )}$

with and   . ${\displaystyle Z_{0}=0}$ ${\displaystyle Z_{n}\geq 0}$

Continue to be given a real number and here for ${\displaystyle \epsilon >0}$ ${\displaystyle i=1,\ldots ,n}$

${\displaystyle B_{i}=\bigcap _{k=0}^{i-1}{\{\omega \in \Omega \mid Z_{k}(\omega )<\epsilon \}}}$

set.

Then:

${\displaystyle \operatorname {P} {\bigl (}\max(Z_{1},\dots ,Z_{n})\geq \epsilon {\bigr )}\leq {\frac {1}{\epsilon }}\cdot \sum _{i=1}^{n}{\int _{B_{i}}(Z_{i}-Z_{i-1})\mathrm {d} {\operatorname {P} }}}$   .

Conclusion: The inequality of Hájek and Rényi

With Frank's lemma, an inequality presented by Jaroslav Hájek and Alfréd Rényi can be derived, which in turn includes further inequalities and in particular both the Kolmogoroff and Chebyshev inequality.

According to Heinz Bauer, the inequality reads as follows:

Be on the probability space finite number of independent integrable real random variables given ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$ ${\displaystyle X_{1},\ldots ,X_{n}\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} \;(n\in \mathbb {N} )}$

plus positive numbers in descending order   . ${\displaystyle {\gamma }_{1}\geq \ldots \geq {\gamma }_{n}>0}$

Be here for ${\displaystyle i=1,\ldots ,n}$

${\displaystyle S_{i}=\sum _{j=1}^{i}{{\bigl (}X_{j}-\operatorname {E} (X_{j}){\bigr )}}}$

set.

Then for every index and for every real one ${\displaystyle m=1,\ldots ,n}$ ${\displaystyle \epsilon >0}$

the inequality

${\displaystyle \operatorname {P} {\bigl (}\max({{\gamma }_{m}{|S_{m}|}},\dots ,{{\gamma }_{n}{|S_{n}|}})\geq \epsilon {\bigr )}\;\leq \;{\frac {{{\gamma }_{m}}^{2}\sum _{j=1}^{m}{\operatorname {Var} (X_{j})}\;+\;\sum _{j=m+1}^{n}{{{\gamma }_{j}}^{2}\operatorname {Var} (X_{j})}}{{\epsilon }^{2}}}}$   .

Fulfills.

Sources and background literature

• Heinz Bauer: Probability Theory and Basics of Measure Theory (=  De Gruyter textbook ). 3rd, revised edition. de Gruyter , Berlin (ua) 1978, ISBN 3-11-007698-5 . MR0936419
• Ove Frank: Generalization of an inequality of Hájek and Rényi . In: Skand. Actuarial writing . tape 49 , 1966, pp. 85-89 . MR0231420
• J. Hájek, A. Rényi: Generalization of an inequality of Kolmogorov . In: Acta Mathematica Academiae Scientiarum Hungaricae . tape 6 , 1955, pp. 281-283 . MR0076207

References and footnotes

1. ↑ ^{a b c} Heinz Bauer: Probability Theory and Basics of Measure Theory. 1978, p. 171 ff.
2. ↑ For an integrable real random variable is the expected value of . ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X)}$ ${\displaystyle X}$
3. ↑ For an integrable real random variable is the variance of . ${\displaystyle X}$ ${\displaystyle \operatorname {Var} (X)}$ ${\displaystyle X}$
4. ↑ A sum of the form is regarded as the sum over the empty set and thus equal to zero. ${\displaystyle \sum _{j=n+1}^{n}a_{j}}$