Level crossings of the empirical process

The asymptotics for the number of times the empirical distribution function crosses the true distribution function are well-known (see Dwass, 1961; or Shorack and Wellner, 1986). We give a process version of this limit theorem and we identify the limiting process to be the local time of Brownian bridge. This substantially strengthens the usual central limit theorem for linear empirical processes. As a by-product of these results, we answer an open problem cited in Shorack and Wellner (1986).