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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|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====
<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>