A L\'evy-Ottaviani type inequality for the Bernoulli process on an interval

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 $(a_i{)}_{i=1}^n$ and for independent Bernoulli random variables $({\epsilon }_i{)}_{i=1}^n$, $$\mathbb{P}$(\sup_{t\in [0,1]}\sum^n_{i=1}a_i(t)\varepsilon_i\geq c)$isdominatedby$C$\mathbb{P}$(\sum^n_{i=1}a_i(1)\varepsilon_i\geq1)$,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 $C=2$.