Lévy inequality

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)