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

Abstract: In this paper we prove a L\'evy-Ottaviani type of property for the Bernoulli process defined on an interval. Namely, we show that under certain conditions on functions $(a_i)<em>{i=1}<sup>{n}$ and for independent Bernoulli random variables $(\varepsilon_i)</em>{i=1}<sup>{n}$, $$\mathbb{P}(\sup_{t\in [0,1]}\sum^{n_{i=1}a_i(t)\varepsilon_i\geq} c)$$ is dominated by $$C\mathbb{P}(\sum<sup>n_{i=1}a_i(1)\varepsilon_i\geq1)$$, 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 $C=2$.