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 $(\varepsilon_i)_{i=1}^{n}$, $\mathbb{P}(\sup_{t\in [0,1]}\sum^n_{i=1}a_i(t)\varepsilon_i\geq c)$ is dominated by $C\mathbb{P}(\sum^n_{i=1}a_i(1)\varepsilon_i\geq1)$, 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 $C=2$.