The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561-574; Schatte, Math. Nachr. 137 (1988) 249-256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular,...

Motivated by problems in functional data analysis, in this paper we prove the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure.

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators and tests. Trimming also provides a profound insight into the partial sum behavior of i.i.d. sequences. There is a wide and nearly complete...

We show that most random walks in the domain of attraction of a symmetric stable law have a non-trivial almost sure central limit theorem with the normal law as the limit.

We obtain a strong approximation for the logarithmic average of sample extremes. The central limit theorem and laws of the iterated logarithm are immediate consequences.

We study the asymptotic behavior of the empirical distribution function and the empirical process of squared residuals. We prove the Glivenko-Cantelli theorem for the empirical distribution function. We show that the two-parameter empirical process converges to a Gaussian process.

A class of iterated processes is studied by proving a joint functional limit theorem for a pair of independent Brownian motions. This Strassen method is applied to prove global (t -- [infinity]), as well as local (t -- 0), LIL type results for various iterated processes. Similar results are also...

Let (W(t), t[greater-or-equal, slanted]0), be a standard Wiener process and define - M+ (t) = max{W(u): u[less-than-or-equals, slant]t},- M-(t) = max{-W(u): u[less-than-or-equals, slant]t},- Z(t) = max{u [less-than-or-equals, slant] t: W(u) = 0}. We investigate the asymptotic behaviour of Z(t)...

Let L(a, t) be the local time of a Wiener process, and put . It is shown that if g(t)=t1/2(log t)-1(log log t)-1 and . A similar result is proved for random g(t) depending on the maximum of the Wiener process. These results settle a problem posed by Csörgo and Révész [7].

We obtain some liminf limits for the Wiener sheet. The approach relies on a careful analysis of the lower tail of the Ornstein-Uhlenbeck process. Our results can be applied to normalized Kiefer and empirical processes. In particular, they yield a satisfying answer to Hirsch's problem for the...