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

In this paper we prove a Levy-Ottaviani type of property for the Bernoulli process dened on an interval. Namely, we show that under certain conditions on functions (ai)ni =1 and for independent Bernoulli random variables ("i)ni=1, P(supt2[0;1]Pni=1 ai(t)"i > c) is dominated by CP(Pni=1 ai(1)"i > 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.