Based on an R2-valued random sample {(yi,xi),1≤i≤n} on the simple linear regression model yi=xiβ+α+εi with unknown error variables εi, least squares processes (LSPs) are introduced in D[0,1] for the unknown slope β and intercept α, as well as for the unknown β when α=0. These LSPs...

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice that is obtained from by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating...

We deduce a partial version of the KMT (1975) inequality for coupling the uniform empirical process with a sequence of Brownian bridges via the construction used by Cs¨org?o and R´ev´esz (CsR) (1978) for their similar coupling of the uniform quantile process with another sequence of Brownian...

It is well known that, asymptotically, the appropriately normalized Vervaat process behaves like one half times the squared empirical process. Considering these two processes as elements of the Lp-space, 1[less-than-or-equals, slant]p[infinity], we give a complete description of the strong and...

In this paper, we first establish a large deviation for increments of a Wiener process. A functional modulus of continuity for a Wiener process is presented and proved using this large deviation.

In this paper we study strong approximations (invariance principles) of the sequential uniform and general Bahadur-Kiefer processes of long-range dependent sequences. We also investigate the strong and weak asymptotic behavior of the sequential Vervaat process, i.e., the integrated sequential...

Let be a non-separable Banach space of real-valued functions endowed with a weighted sup-norm. We consider partial sum processes as random functions with values in . We establish weak convergence statements for these processes via their weighted approximation in probability by an appropriate...