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In this paper, we give necessary and sufficient conditions for a stationary sequence of random variables with values in a separable Hilbert space to satisfy the conditional central limit theorem introduced in Dedecker and Merlevede (Ann. Probab. 30 (2002) 1044–1081). As a consequence, this theorem implies stable convergence of the normalized partial sums to a mixture of normal distributions. We also establish the functional version of this theorem. Next, we show that these conditions are satisfied for a large class of weakly dependent sequences, including strongly mixing sequences as well as mixingales. Finally, we present an application to linear processes generated by some stationary sequences of H -valued random variables.

Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.

Weak invariance principles are established for strictly stationary weakly dependent sequences, having a decomposed strong mixing coefficient into two parts, one based on the strong mixing condition with a polynomial mixing rate and other based on the ρ-mixing condition. The result is applied to the output of the Tukey ‘3R smoother’.

We compare three dependence coefficients expressed in terms of conditional expectations, and we study their behaviour in various situations. Next, we give a new covariance inequality involving the weakest of those coefficients, and we compare this bound to that obtained by Rio (Ann. Inst. H. Poincare Probab. Statist. 29 (1993) 587–597) in the strongly mixing case. This new inequality is used to derive sharp limit theorems, such as Donsker's invariance principle and Marcinkiewicz's strong law. As a consequence of a Burkholder-type inequality, we obtain a deviation inequality for partial sums.

We count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a "local time," whose inverse is a stable subordinator of a known index, and a Brownian motion. The study of this limit process leads to some precise sample path properties of Brownian motion. These properties improve earlier results of Dvoretsky and Kahane on the existence of small oscillations in the Brownian path.