We consider the empirical two-sample U-statistic with $$\beta$$ -mixing strictly stationary data and investigate its convergence in Skorohod spaces. We then provide an application of such convergence.

We present a robust test for change-points in time series which is based on the two-sample Hodges-Lehmann estimator. We develop new limit theory for a class of statistics based on the two-sample U-quantile processes, in the case of short range dependent observations. Using this theory we can derive the asymptotic distribution of our test statistic under the null hypothesis. We study the finite sample properties of our test via a simulation study and compare the test with the classical CUSUM test and a test based on the Wilcoxon-Mann-Whitney statistic.

Motivated by some common-change point tests, we investigate the asymptotic distribution of the U-statistic process U_n(t)=∑_i=1^[nt]∑_j=[nt]+1^n h(X_i,X_j), 0≤ t≤ 1, when the underlying data are long-range dependent. We present two approaches, one based on an expansion of the kernel h(x,y) into Hermite polynomials, the other based on an empirical process representation of the U-statistic. Together, the two approaches cover a wide range of kernels, including all kernels commonly used in applications.

On July 19, 2006, Professor Walter Philipp passed away during a hike in the Austrian Alps as a result of a sudden heart attack. By the time of his death, Walter Philipp had been for almost 40 years on the faculty of the University of Illinois at Urbana-Champaign, for the last couple of years as professor emeritus. He is survived by his wife Ariane and his four children, Petra, Robert, Anthony and André. Walter Philipp is sorely missed by his family, but also by his many colleagues, coauthors and former students all over the world, to whom he was a loyal and caring friend for a long time, in some cases for several decades. Walter Philipp was born on December 14, 1936 in Vienna, Austria, where he grew up and lived for most of the first 30 years of his life. He studied mathematics and physics at the University of Vienna, where he obtained his Ph.D. in 1960 and his habilitation in 1967, both in mathematics. From 1961 until 1967 he was scientific assistant at the University of Vienna. During this period, Walter Philipp spent two years as a postdoc in the US, at the University of Montana in Missoula and at the University of Illinois. In the fall of 1967 he joined the faculty of the University of Illinois at Urbana-Champaign, where he would stay for the rest of his life. Initially, Walter Philipp was on the faculty of the Mathematics Department, but in 1984 he joined the newly created Department of Statistics at the University of Illinois. From 1990 until 1995 he was chairman of this Department. While on sabbatical leave from the University of Illinois, Walter spent longer periods at the University of North Carolina at Chapel Hill, at MIT, at Tufts University, at the University of Göttingen and at Imperial College, London. Walter Philipp received numerous recognitions for his work. Most outstanding of these was his election to membership in the Austrian Academy of Sciences. As a student and postdoc at the University of Vienna, Walter Philipp worked under the guidance of Professor Edmund Hlawka, founder of the famous postwar Austrian school of analysis and number theory. Other former students of Professor Hlawka include Johann Cigler, Harald Niederreiter, Wolfgang M. Schmidt, Fritz Schweiger and Robert Tichy. It was here that Walter Philipp got in touch with the classical topics from analysis and number theory that would guide a large part of his research for the rest of his life. Uniform distribution, discrepancy of sequences, number-theoretic transformations associated with various expansions of real numbers, additive number-theoretic functions, Diophantine approximation, lacunary series, became recurrent themes in Walter Philipp’s subsequent work. He studied these themes using techniques from probability theory, e.g. for mixing processes, martingales and empirical processes. He contributed greatly to the development of several branches of probability theory and solved much investigated, difficult problems in analysis and number theory with the help of the tools he developed.

In this paper we study the conditional distribution of a multinomial sample, given certain marginals. By direct evaluation of the probabilities using Stirling's formula we obtain a local limit theorem. The limit distribution turns out to be multivariate normal and we can identify the covariance matrix. Consequently we prove asymptotic normality of the H-coefficient in certain nonparametric unfolding models with dichotomous data.

We present a new approach to the problem of time series modelling that captures the invariant distribution of time series data within the model. This is particularly relevant in modelling economic and financial time series, such as oil prices, that exhibit clustering around a few preferred market modes. We propose a potential function approach which determines the function that governs the underlying time series process. This approach extends naturally to modelling multivariate time series. We show how to estimate the potential function for dimensions one and higher and use it to model statistically the evolution of the time series. An illustration of the procedure shows that testing the resulting model against historical data of oil prices captures the essential price behavior remarkably well. The model allows the generation of copies of the observed time series as well as providing better predictions by reducing uncertainty about the future behavior of the time series.

Motivated by the problem of estimation of the fractal dimension from a nonlinear time series, we study the behavior of U-statistics for dependent observations. We give a review of U-statistics laws of large numbers and outline main methods and techniques used in obtaining these results. We present a new mixing condition of near regularity, which adequately describes dependence in processes arising from weak Bernoulli dynamical systems. With the. help of this notion we obtain such asymptotic results as U-statistics strong law of large numbers and weak convergence of empirical processes of U-statistics structure.

In this paper we develop a general approach for investigating the asymptotic distribution of functionals X-n = f((Z(n+k))(k is an element ofZ)) of absolutely regular stochastic processes (Z(n))(n is an element ofZ). Such functionals 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[GRAPHICS]with symmetric kernel h : R x 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 (U-n(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.

Motivated by the problem of estimating the fractal dimension of a strange attractor, we prove weak consistency of U-statistics for stationary ergodic and mixing sequences when the kernel function is unbounded, extending by this earlier results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss. We apply the obtained results to show consistency of the Takens estimator for the con-elation dimension.

We address the problem of prediction of nonlinear time series by kernel estimation of autoregression, and introduce a variation of this method. We apply this method to an experimental time series and compare its performance with predictions by feed-forward neural networks as well as with fitting a local and a global linear autoregression.

Consider a random integer-valued process X(t) on Z(+) that satisfies some weak dependence condition. We study the empirical distribution function of the occupation times of such a process and prove convergence to a suitable Gaussian process. An application to the statistical analysis of open and closed sojourn-time distributions for ion channels is provided.

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.

We study the bootstrap distribution for U-statistics with special emphasis on the degenerate case. For the Efron bootstrap we give a short proof of the consistency using Mallows′ metrics. We also study the i.i.d. weighted bootstrap [formula] where (Xi) and (ξi) are two i.i.d. sequences, independent of each other and where Eξi = 0, Var(ξi) = 1. It turns out that, conditionally given (Xi), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral ∫10 ∫10h(F−1(x), F−1(y)) dW(x) dW(y). As a by-product we get an a.s. limit theorem for the eigenvalues of the matrix Hn=((1/n)h(Xi, Xj))1 ≤ i, j ≤ n.

The transformation of radiocarbon years to calendar years (cal AD/BC) is not straightforward because of past variations in atmospheric 14C content (de Vries 1958; Suess 1970). A calibration curve, y = f(x), transforms each dendrochronologically dated calendar age (x) to a 14C date (y). By inverting this relationship, one can determine the calibrated calendar age of a given sample. In some time intervals, the calibration curve is problematic in that f(x) is not uniquely invertible (Fig. 1); even an exact measurement of y cannot be converted to a single calendar age (see examples in van der Plicht & Mook (1987)).

Motivated by recent results on pathwise central limit theorems, we study in a systematic way log-average versions of classical limit theorems. For partial sums S(k) of independent r.v.'s we prove under mild technical conditions that (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)I{S(k)/a(k) is-an-element-of .} --> G(.) (a.s.) if and only if (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)P(S(k)/a(k) is-an-element-of .) --> G(.). A functional version of this result also holds. For partial sums of i.i.d. r.v.'s attracted to a stable law, we obtain a pathwise version of the stable limit theorem as well as a strong approximation by a stable process on log dense sets of integers. We also give necessary and sufficient conditions for the law of large numbers in log density.

Let (Xj)j∞=1 be a stationary, mean-zero Gaussian sequence with covariances r(k)=EXk+1X1 satisfying r(0)=1 and r(k)=k−DL(k) where D is small and L is slowly varying at infinity. Consider the sequence Yj=G(Xj),j=1,2,..., where G is any measurable function. We obtain the asymptotic distribution of certain degenerate von Mises and U-statistics based on theYj. As applications, we consider the sample variance, the χ2-squared goodness of fit test and the Cramer-von Mises-Smirnov ω2 criterion. The results are non-standard.

It is well known that symmetric statistics based on a kernel with finite second moment have a limit law which can be described by a multiple Wiener-Ito integral. However, if the kernel has less than second moments, no weak limit law holds in general. In the present paper we show that by a suitable change of the empirical process this process has a p-stable multiple integral as its limit.