Cantelli's inequality

The inequality of Cantelli is an elementary stochastic inequality that the Italian mathematician Francesco Paolo Cantelli back. It is related to the Chebyshev-Markov inequality and provides a one-sided estimate of the probability that a real random variable will exceed its expected value by a positive number .

Formulation of the inequality

The Cantellian inequality can be stated as follows:

A probability space and a real random variable are given   . ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$ ${\displaystyle X\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} }$

${\displaystyle X}$ have a finite second moment :

${\displaystyle \operatorname {E} (X^{2})<{\infty }}$   .

Let a real number be     given and denote the variance of . ${\displaystyle c>0}$ ${\displaystyle \operatorname {V} (X)}$ ${\displaystyle X}$

Then there is the inequality

${\displaystyle \operatorname {P} {{\bigl (}X\geq {\operatorname {E} {(X)}+c}{\bigr )}}\leq {\frac {\operatorname {V} {{\bigl (}X{\bigr )}}}{c^{2}+{\operatorname {V} (X)}}}}$   .

Proof of the inequality

Following the presentation by Klaus D. Schmidt , it can be derived as follows:

Step 1

One sets

${\displaystyle Z=X-{\operatorname {E} (X)}}$ .  

Then first is

${\displaystyle \operatorname {E} (Z)=0}$

and further

${\displaystyle \operatorname {V} (Z)=\operatorname {V} (X)=\operatorname {E} {\bigl (}Z^{2}{\bigr )}}$ .  

step 2

If one now has a (initially arbitrary) real number , the following chain of inequalities results, especially because of the Chebyshev-Markov inequality for second moments : ${\displaystyle t>-c}$

${\displaystyle {\begin{aligned}\operatorname {P} {{\bigl (}X\geq {\operatorname {E} {\bigl (}X{\bigr )}+c}{\bigr )}}&=\operatorname {P} {\bigl (}Z\geq c{\bigr )}\\&=\operatorname {P} {\bigl (}Z+t\geq c+t{\bigr )}\\&\leq \operatorname {P} {\bigl (}|Z+t|\geq c+t{\bigr )}\\&\leq {\frac {\operatorname {E} {\bigl (}{(Z+t)}^{2}{\bigr )}}{{(c+t)}^{2}}}\\&={\frac {\operatorname {E} {\bigl (}Z^{2}{\bigr )}+t^{2}}{{(c+t)}^{2}}}\\&={\frac {{\operatorname {V} (Z)}+t^{2}}{(c+t)^{2}}}\\&={\frac {{\operatorname {V} (X)}+t^{2}}{(c+t)^{2}}}.\end{aligned}}}$

step 3

Especially for the real number

${\displaystyle t_{0}={\frac {\operatorname {V} (X)}{c}}}$

applies after step 2:

${\displaystyle {\begin{aligned}\operatorname {P} {{\bigl (}X\geq {\operatorname {E} (X)+c}{\bigr )}}&\leq {\frac {{\operatorname {V} (X)}+{t_{0}}^{2}}{(c+{t_{0}})^{2}}}\\&={\frac {{\operatorname {V} (X)}+{\frac {{\operatorname {V} (X)}^{2}}{c^{2}}}}{{(c+{\frac {\operatorname {V} (X)}{c}})}^{2}}}\\&={\frac {{c^{2}}\cdot {\operatorname {V} (X)}+{{\operatorname {V} (X)}^{2}}}{{{c^{2}}\cdot (c+{\frac {\operatorname {V} (X)}{c}})}^{2}}}\\&={\frac {{\operatorname {V} (X)}\cdot {{\bigl (}{c^{2}}+{{\operatorname {V} (X)}{\bigr )}}}}{{({c^{2}}+{\operatorname {V} (X)})}^{2}}}\\&={\frac {\operatorname {V} {{\bigl (}X{\bigr )}}}{c^{2}+{\operatorname {V} (X)}}}\\\end{aligned}}}$   .

So everything is proven.

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The real-valued function occurring in step 2 above

${\displaystyle t\mapsto {\frac {{\operatorname {V} (X)}+t^{2}}{(c+t)^{2}}}\;(t\in \;]-c\;,\;\infty [)}$

takes at the named place

${\displaystyle t_{0}={\frac {\operatorname {V} (X)}{c}}}$

their bare minimum . The upper bound mentioned in the Cantellian inequality is therefore optimal in this sense.

swell

• Klaus D. Schmidt: Measure and probability (=  Springer textbook ). Springer Verlag, Berlin, Heidelberg 2009, ISBN 978-3-540-89729-3 .

References and footnotes

1. ↑ Klaus D. Schmidt: Measure and probability. 2009, pp. 288-289
2. ↑ For a real random variable , its expected value is designated. ${\displaystyle \xi }$ ${\displaystyle \operatorname {E} (\xi )}$
3. ↑ For a real random variable , its variance is designated. ${\displaystyle \xi }$ ${\displaystyle \operatorname {V} (\xi )}$