The purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid Burgers equation having a typical realizationb(y) of the Brownian motion as an initial datum. This case was proposed and studied numerically in the companion paper by She, Aurell and Frisch. The description of the statistics is given in terms of the behavior of the convex hull of the random process\(w(y) = \int\limits_0^y {(b(\eta ) + \eta )} d\eta \). The Hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied.