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In this paper we extend the contraction mapping method to prove various existence and uniqueness properties of (self-similar) random fractal measures, and establish exponential convergence results for approximating sequences defined by means of the scaling operator. For this purpose we introduce a version of the Monge Kantorovich metric on the class of probability distributions of random measures in order to prove the relevant results in distribution. We also use a special sample space of “construction trees” on which we define the approximating sequence of random measures, and introduce a certain operator and a compound variant of the Monge Kantorovich metric in order to establish a.s. exponential convergence to the unique random fractal measure. The arguments used apply at the random measure and random measure distribution levels, and the results cannot be obtained by previous contraction arguments which applied at the individual realisation level.

Let (X i, A i, P i), i = 1, 2 be two probability spaces. A probability µ on (X 1 × X 2, A 1⊗A 2) is said to have marginals P 1 and P 2 if $$P(A_1 \times X_2) = P_1 (A_1) \textup{for all} A_1 \in \mathcal {A}_1 \\ \textup{and} \\ P(X_1 \times A_2) = P_2 (A_2) \textup{for all} A_2 \in \mathcal {A}_2 \\$$ Let \(\mathcal{M} = \left\{\mu \ \textup{on} \ \mathcal{A}_1 \otimes \mathcal{A}_2: \mu\ \textup a probability with marginals \ P_1 \ \textup{and} \ P_2 \right\}\).

As consequence of a characterization of optimal multivariate coupling (transportation) problems we obtain the existence of optimal Monge solutions as well as an explicit construction method for optimal transportation plans in the case that one mass distribution is discrete. We also give a new characterization of an extension of the transportation problem with more than two mass distributions involved.

Im folgenden sollen Probleme und Methoden der Stochastik anhand einiger Beispiele erlautert werden. Ich werde dabei nicht so sehr auf die innermathematische Entwicklung dieses Gebietes eingehen, sondern die Aspekte betonen, die von interdisziplinarem Charakter sind.

Several examples of the asymptotic analysis of recursive algorithms are investigated by the contraction method. The examples concern random permutations and binary search trees. In these examples it is demonstrated that the contraction method can be applied successfully to problems with contraction constants converging to one and with nonregular normalizations as logarithmic normalizations, which are typical in search type algorithms. An advantage of this approach is its generality and the possibility to obtain quantitative approximation results.

The iterative proportional fitting procedure (IPFP) was introduced in 1940 by Deming and Stephan to estimate cell probabilities in contingency tables subject to certain marginal constraints. Its convergence and statistical properties have been investigated since then by several authors and by several different methods. A natural extension of the IPFP to the case of bivariate densities has been introduced by Ireland and Kullback. It has been conjectured that also in the general case the IPFP converges to the minimum discrimination projection on the class of distributions with given marginals. We verify this conjecture under some regularity conditions.

In this paper, the rate of convergence in the CLT is estimated w.r.t. the Kantorovich metric for random variables with values in separable Banach spaces. In the first part, the rate in stable limit theorems for sums of i.i.d. random variables is considered. The method of proof is an extension of the Bergstrom convolution method. All assumptions regarding the domain of attraction are given in a metric form. In the second part of the paper an extension is given to the martingale case. For the proof we investigate smoothing properties of suitable conditional versions of the Kantorovich metric. As a consequence of the results for the Kantorovich metric one obtains rate of convergence results in stable limit theorems for martingales w.r.t. the Prohorov metric.

We construct a.s. nonlinear regression representations of general stochastic processes (X n ) nϵ N . As a consequence we obtain in particular special regression representations of Markov chains and of certain m-dependent sequences. For m-dependent sequences we obtain a constructive method to check, whether these sequences have a monotone (m+1)-block factor representation.

Let $\zeta$ be chosen at random on the surface of the $p$-sphere in $\mathbb{R}^n, 0_{p,n} := \{x \in \mathbb{R}^n: \sum^n_{i = 1}|x_i|^p = n\}$. If $p = 2$, then the first $k$ components $\zeta_1,\ldots, \zeta_k$ are, for $k$ fixed, in the limit as $n\rightarrow\infty$ independent standard normal. Considering the general case $p > 0$, the same phenomenon appears with a distribution $F_p$ in an exponential class $\mathscr{E}. F_p$ can be characterized by the distribution of quotients of sums, by conditional distributions and by a maximum entropy condition. These characterizations have some interesting stability properties. Some discrete versions of this problem and some applications to de Finetti-type theorems are discussed.

This paper gives a review of Frechet-bounds and their applications. In section two an approach to the marginal problem and Frechet-bounds based on duality theory resp. the Hahn-Banach theorem is discussed. Main applications concern the Strassen representation theorem for stochastic orders, the sharpness of the classical Frechet-bounds, the representation of minimal metrics, couplings of distributions, the Monge-Kantorovic-problem, the construction of random variables with maximum (resp. minimum) sum and variances of the sum, maximally dependent random variables and others. For multivariate marginal systems there is a useful reduction principle and there are some bounds for simple systems, which yield a characterization of the marginal problem for a system of two dimensional marginals in a three-fold product space. In section three we discuss some generalizations of the Younginequality, which are useful for solving the dual problems of the Frechet-bounds. A basic notion in this connection is the notion of c-convex functions. As an application one can give a nice characterization of solutions of certain transportation problems. We give a probabilistic proof of some generalizations of the Young- and the Oppenheim-inequality. In section four we discuss some statistical applications and problems. The Huzurbazar conjecture on marginal sufficiency, the problem of the optimal combination of marginal tests and the question of estimation theory in marginal models is considered.

In the present paper we deal with the characterization of some dependence concepts for the multivariate normal distribution. It turns out that normal distributions have some special properties w.r.t. these dependence concepts and, furthermore, that the characterizations are closely connected to some interesting problems on matrices. Some applications to simultaneous confidence bounds are discussed.

Let X (1) ,…, X ( n ) be the order statistics of n iid distributed random variables. We prove that ( X ( i ) ) have a certain Markov property for general distributions and secondly that the order statistics have monotone conditional regression dependence. Both properties are well known in the case of continuous distributions.

We make some remarks on the problem how to construct probability measures with given marginals. Questions of this kind arise if one wants to build a stochastic model in a situation where one has some idea of the kind of dependence and knows exactly certain marginal distributions.

Motivated by a problem in PERT networks we consider the question of construction of random variables with maximum sums when the marginals are fixed.

It is shown that the coupling of distributions by their quantile transforms leads to simple and unifying proofs of monotonicity and unbiasedness properties of tests in many nonparametric testing problems

The first part of the present paper is concerned with closedness properties of subsets of monotone test-functions on Rn w.r.t. weak *-topology. In the second part of this paper these results are applied to derive some results on essential completeness of monotone tests and estimators. Especially, some results of OOSTERHOFF and BROWN. COHEN, STRAWDERMAN on monotone tests are generalized and their sufficient conditions are weakened.

The dependence concept of weak association is introduced and is shown to be equivalent to positive quadrant dependence. Furthermore, a characterization of independence in the class of positive quadrant dependent random variables by means of moment conditions is proved. Both results generalize some theorems proved by Lehmann and Jogdeo for the two- and three-dimensional case.

This paper investigates the distribution-free outer confidence interval for the quantile interval introduced by Wilks. Besides exact expressions and a recurrence formula some bounds are derived for the probability of correct coverage of the quantile interval which improve bounds known up to now, and which are tested in several examples. Asymptotic approximations are discussed and applications of the developed methods to other statistics based on order statistics are indicated.

We characterize the projection of a probability measure w.r.t. divergence type distances on a class of distributions determind by inequality constraints. We discuss in more detail the KULLBACK-LEIBLER- and the distances.

Inequalities for the rearrangement of functions are applied to obtain a solution of a statistical optimization problem. This optimization problem arises in situations where one wants to describe the influence of stochastic dependence on a statistical problem.

Some characterizations for the stochastic ordering of probability distributions are given. Especially a general sufficient condition for stochastic ordering is proved and the question of existence of upper and lower bounds in the class of all distributions with given marginals is considered. Stochastic ordering is applied to prove a general theorem on monotonicity of the OC-function of sequential probability ratio tests in stochastic processes.

For some subsets of the set of all Δ-monotone functions on [0,1] n we characterize distribution functions F, G such that E F f≦EG f for all f within these subsets. Furthermore, we determine sharp upper and lower bounds of integrals of functions in these subsets w.r.t. all distributions with fixed marginals and give some applications of these results.

By an extension of the idea of the multivariate quantile transform we obtain an explicit formula for the Wasserstein distance between multivariate distributions in certain cases. For the general case we use a modification of the definition of the Wasserstein distance and determine optimal ‘markov-constructions’. We give some applications to the problem of approximation of stochastic processes by simpler ones, as e.g. weakly dependent processes by independent sequences and, finally, determine the optimal martingale approximation to a given sequence of random variables; the Doob decomposition gives only the ‘one-step optimal’ approximation.

We construct minimum variance unbiased estimators of von Mises functionals in estimation problems where no complete sufficient σ-algebra exists. The construction method is instead based on the higher order tangent structure of the underlying class of distributions. We discuss especially some curved and some noncurved nonparametric examples in the iid case and estimation in nonparametric Markov chain models.

We introduce a nonparametric location model and discuss its relations to the more familiar parametric location model and the mixture location model. The model can be characterized as a random translation model. Further characterizations are given by means of conditional distributions, by characteristic functions and by the local structure of the model in terms of the tangent cone. The relations between two different nonparametric location models can be made explicit. We consider especially the total variation distance and the deficiency. Consequences of these results for the statistical analysis of models of this type will be discussed in a subsequent paper.

For a class of point processes including nonhomogeneous Poisson processes, some Cox processes and some processes of non-Poisson type, the question of existence and construction of complete and sufficient statistics is investigated. The results are based on the generalization of some results from sequential analysis and are applied to the construction of optimal tests and estimators.

Two examples for marked point processes are discussed and some characteristic parameters of these models are calculated. Both examples are in some way modifications of the counter models which are well known and treated in several textbooks. MARKED POINT PROCESS; COUNTER MODEL; RENEWAL PROCESS

Some recent results of Oxley and Welsh (1979) and McDiarmid (1981) concerning Bernoulli percolation on clutters are generalized. Our results allow to consider quantitative aspects of percolation on graphs and clutters. PERCOLATION PROBABILITY; CLUTTER; RANDOM GRAPH; ASSOCIATION

A convergence theorem of Billingsley for the empirical process of stationary, real valued radom variables under a mixing condition is generalized to the k -dimensional and nonstationary case. Further a more general empirical process is treated, including the upper summation boundary as argument. Some applications are given to a Kolmogorov-Smirnov and a Cramer-von Mises type statistic.

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.

Same explicit optimal coupling results are derived with respect to minimal metrics of lp-type. In particular, the optimality of radial transformations, positive transformations, and monotone transformations of the components is established.

We study random algorithms arising in multiple access communication problems. We prove asymptotic stability and normality. Numerical analysis of the performance of the algorithms is provided. The general convergence theorems in the paper are based on contraction properties of suitably chosen (ideal) metrics. The approach allows us to prove asymptotic normality under very weak conditions, superseding the results of other authors. Stable and multivariate extensions seem to be analysed for the first time in the literature. Our numerical results show that the Capetanakis—Tsybakov—Mikhailov (CTM) algorithm and the trinomial algorithm have a similar asymptotic behaviour. For a small number of users there are some differences concerning the quality of the normal approximation.

In this paper we obtain based on an idea of M. Knott and C. S. Smith (1994, Linear Algebra Appl.199, 363–371) characterizations of solutions of three-coupling problems by reduction to the construction of optimal couplings of each of the variables to the sum. In the case of normal distributions this leads to a complete solution. Under a technical condition this idea also works for general distributions and one obtains explicit results. We extend these results to the n-coupling problem and derive a characterization of optimal n-couplings by several 2-coupling problems. This leads to some constructive existence results for Monge solutions.

On utilise la polarisation pour unifier et etendre les resultats classiques concernant la completude des statistiques d'ordre generalisees avec des distributions dominees et non dominees

Untersucht wird das Konvergenz- und Konsistenzverhalten von Tests und Schatzern. Die Begriffe Konvergenz und Konsistenz lassen sich dabei im asymptotischen Modell formulieren.

By means of the Poisson type representation of multivariate empirical processes and using a generalization of the Birnbaum Marshall inequality it is shown that the empirical process converges in distribution even when it is weighted by some unbounded weighting functions.

By means of a general weak convergence theorem some invariance principles are proven for the multivariate sequential empirical process and for the multivariate rank order process w.r.t. stronger metrics than the generalized Skorohod metric. The underlying random variables are neither assumed to be independent nor to be stationary. These results are then applied to derive convergence of the weighted empirical cumulatives and for the weighted rank order process. Finally by a new representation asymptotic normality is proven for a general class of linear multivariate rank order statistics.

Die wesentliche Idee der lokalen asymptotischen Statistik ist es, ein statistisches Experiment umzuparametrisieren und damit in eine ‘konvergente’ Folge von Experimenten einzubetten. Konvergenz bedeutet dabei die Konvergenz der Likel ihood-Quotientenprozesse. Der klassische Weg der asymptotischen Analyse solcher Experimente basiert auf der Approximation der Likelihoodprozesse durch die Likelihoodprozesse von statistisch wohlbekannten Experimenten — den Exponentialfamilien — und einer Ruckfuhrung auf die Analyse von Exponentialfamilien. Dieser Weg soll hier im folgenden im wesentlichen dargestellt werden.