Herold Dehling

Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation

In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

Large deviations for some weakly dependent random processes

In this paper we compute large deviation probabilities for two classes of weakly dependent processes, moving averages of i.i.d. random variables and Poisson center cluster random measures.

Limit Theorems for Sums of Weakly Dependent Banach Space Valued Random Variables

SummaryWe prove an estimate for the Prohorov-distance in the central limit theorem for strong mixing Banach space valued random variables. Using a recent variant of an approximation theorem of Berkes and Philipp (1979) we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term

Strong laws for L- and U-statistics

t^{\tfrac{1}{2} - \gamma }

Central limit theorem and the bootstrap for U-statistics of strongly mixing data

. For strong mixing sequences we prove a strong invariance principle with error term o((t log logt)1/2).

Stochastic models for transport in a fluidized bed

We propose a new test against a change in correlation at an unknown point in time based on cumulated sums of empirical correlations. The test does not require that inputs are independent and identically distributed under the null. We derive its limiting null distribution using a new functional delta method argument, provide a formula for its local power for particular types of structural changes, give some Monte Carlo evidence on its finite-sample behavior, and apply it to recent stock returns.

Invariance principles for von Mises and U-statistics

Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems; of Hoeffding and of Helmers for lid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.

BIVARIATE SYMMETRICAL STATISTICS OF LONG-RANGE DEPENDENT OBSERVATIONS

The positive dependence notion of association for collections of random variables is generalized to that of weak association for collections of vector valued random elements in such a way as to allow negative dependencies in individual random elements. An invariance principle is stated and proven for a stationary, weakly associated sequence of d-valued or separable Hilbert space valued random elements which satisfy a covariance summability condition.