Some asymptotic properties of the hybrids of empirical and partial-sum processes

The motivation of this paper is to study some properties of the local times (when it exists) of the hybrids of empirical and partial-sum processes defined by \bar{A}_n(t)=\sum_{1\leq i \leq n} H(X_i)1_{\{X_i\leq t\}} \epsilon_i, \quad - \infty namely by using knowing results on empirical process and Brownian local times.