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 $C=2$.
Categories: math.PR