Namespaces
Variants
Actions

Difference between revisions of "Lévy inequality"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(latex details)
 
(3 intermediate revisions by one other user not shown)
Line 1: Line 1:
An inequality for the distribution of the maximum of sums of independent random variables, centred around the corresponding medians. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582901.png" /> be independent random variables, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582902.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582903.png" /> be the median (cf. [[Median (in statistics)|Median (in statistics)]]) of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582904.png" />; then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582905.png" /> one has the Lévy inequalities
+
<!--
 +
l0582901.png
 +
$#A+1 = 10 n = 0
 +
$#C+1 = 10 : ~/encyclopedia/old_files/data/L058/L.0508290 L\Aeevy inequality
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582906.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
An inequality for the distribution of the maximum of sums of independent random variables, centred around the corresponding medians. Let  $  X _ {1} \dots X _ {n} $
 +
be independent random variables, let  $  S _ {k} = \sum_{i=1}^ {k} X _ {i} $
 +
and let  $  mX $
 +
be the median (cf. [[Median (in statistics)]]) of the random variable  $  X $;  
 +
then for any  $  x $
 +
one has the Lévy inequalities
 +
 
 +
$$
 +
{\mathsf P} \left \{ \max _ {1\leq  k \leq  n } ( S _ {k} - m ( S _ {k} -
 +
S _ {n} ) ) \geq  x \right \}  \leq  2 {\mathsf P} \{ S _ {n} \geq  x \}
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582907.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \left \{ \max _ {1\leq  k \leq  n }  | S _ {k} - m ( S _ {k} - S _ {n} ) | \geq  x \right \}  \leq  2 {\mathsf P} \{ | S _ {n} | \geq  x \} .
 +
$$
  
Immediate consequences of these inequalities are the Lévy inequalities for symmetrically-distributed random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582908.png" />:
+
Immediate consequences of these inequalities are the Lévy inequalities for symmetrically-distributed random variables $  X _ {1} \dots X _ {n} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l0582909.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \left \{ \max _ {1 \leq  k \leq  n }  S _ {k} \geq  x \right
 +
\}  \leq  2 {\mathsf P} \{ S _ {n} \geq  x \}
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058290/l05829010.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \left \{ \max _ {1 \leq  k \leq  n }  | S _ {k} | \geq  x
 +
\right \}  \leq  2 {\mathsf P} \{ | S _ {n} | \geq  x \} .
 +
$$
  
 
The Lévy inequalities can be regarded as generalizations of the [[Kolmogorov inequality|Kolmogorov inequality]]. The Lévy inequalities were obtained by P. Lévy [[#References|[1]]] in the investigation of general problems on the convergence of distributions of sums of independent random variables to stable laws. There is also a generalization of them to martingales [[#References|[2]]].
 
The Lévy inequalities can be regarded as generalizations of the [[Kolmogorov inequality|Kolmogorov inequality]]. The Lévy inequalities were obtained by P. Lévy [[#References|[1]]] in the investigation of general problems on the convergence of distributions of sums of independent random variables to stable laws. There is also a generalization of them to martingales [[#References|[2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Lévy,   "Théorie de l'addition des variables aléatoires" , Gauthier-Villars  (1937)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Loève,   "Probability theory" , Princeton Univ. Press  (1963)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> P. Lévy, "Théorie de l'addition des variables aléatoires" , Gauthier-Villars  (1937)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> M. Loève, "Probability theory" , Princeton Univ. Press  (1963)</TD></TR>
 +
</table>

Latest revision as of 10:43, 20 January 2024


An inequality for the distribution of the maximum of sums of independent random variables, centred around the corresponding medians. Let $ X _ {1} \dots X _ {n} $ be independent random variables, let $ S _ {k} = \sum_{i=1}^ {k} X _ {i} $ and let $ mX $ be the median (cf. Median (in statistics)) of the random variable $ X $; then for any $ x $ one has the Lévy inequalities

$$ {\mathsf P} \left \{ \max _ {1\leq k \leq n } ( S _ {k} - m ( S _ {k} - S _ {n} ) ) \geq x \right \} \leq 2 {\mathsf P} \{ S _ {n} \geq x \} $$

and

$$ {\mathsf P} \left \{ \max _ {1\leq k \leq n } | S _ {k} - m ( S _ {k} - S _ {n} ) | \geq x \right \} \leq 2 {\mathsf P} \{ | S _ {n} | \geq x \} . $$

Immediate consequences of these inequalities are the Lévy inequalities for symmetrically-distributed random variables $ X _ {1} \dots X _ {n} $:

$$ {\mathsf P} \left \{ \max _ {1 \leq k \leq n } S _ {k} \geq x \right \} \leq 2 {\mathsf P} \{ S _ {n} \geq x \} $$

and

$$ {\mathsf P} \left \{ \max _ {1 \leq k \leq n } | S _ {k} | \geq x \right \} \leq 2 {\mathsf P} \{ | S _ {n} | \geq x \} . $$

The Lévy inequalities can be regarded as generalizations of the Kolmogorov inequality. The Lévy inequalities were obtained by P. Lévy [1] in the investigation of general problems on the convergence of distributions of sums of independent random variables to stable laws. There is also a generalization of them to martingales [2].

References

[1] P. Lévy, "Théorie de l'addition des variables aléatoires" , Gauthier-Villars (1937)
[2] M. Loève, "Probability theory" , Princeton Univ. Press (1963)
How to Cite This Entry:
Lévy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy_inequality&oldid=17900
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article