A Lévy-Ottaviani type inequality for the Bernoulli process on an interval

Abstract

In this paper we prove a Lévy-Ottaviani type of property for the Bernoulli process defined on an interval. Namely, we show that under certain conditions on functions $(a_i{)}_{i=1}^n$ and for independent Bernoulli random variables $({\epsilon }_i{)}_{i=1}^n$, $\mathbb{P}({\sup }_{t\in [0,1]}{\sum }_{i=1}^n{a}_i(t){\epsilon }_i\ge c)$ is dominated by $C\mathbb{P}({\sum }_{i=1}^n{a}_i(1){\epsilon }_i\ge 1)$, where $c$ and $C$ are explicit numerical constants independent of $n$. The result is a partial answer to the conjecture of W. Szatzschneider that the domination holds with $c=1$ and $C=2$.

6 pages