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The change-of-variance function is defined for estimators of regression coefficients. Both an unstandardized and a standardized form of the change-of-variance sensitivity are introduced, and their relation with the corresponding gross-error-sensitivities is investigated. The problems of optimal robustness lead to the Hampel-Krasker and the Krasker-Welsch estimators. At the same time, also the scale parameter has to be estimated robustly. By means of the change-of-variance sensitivity, optimal robust redescending scale estimators are constructed.

Gaussian fields are considered as Gibbsian fields. Thermodynamic functions are calculated for them and the variational principle is proved. As an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for Gaussian Markov fields.

A hyperfinite filtered Loeb space is a very rich probability space constructed using non-standard analysis. Let b be a given Brownian motion on such a Loeb space. In this paper the infinitesimal techniques of Keisler (An infinitesimal approach to stochastic analysis, to appear) are extended to obtain solutions on this space to equations of the form dyt=f(t,y)dt+g(t,y)dbt te[0,1] where f,g are measurable and f(t,x), g(t,x) depend on the past (xs)s≦t of xeC[0,1].

The purpose of this paper is to apply convex analysis methods to prove existence results for optimal stopping time problems for the diffusion processes considered by Stroock and Varadhan. The work is based on the characterization given by Rost of the measures associated to the stopping times. The method is applied to the problem of control of diffusions where the stopping time is also a control.

We formulate an abstract functional-analytic framework for the study of Gibbs measures on infinite product spaces. Working in this frame-work, we present a detailed analysis of the weak-coupling regime. Specifically, we derive general theorems on existence of the Gibbs measure, analyticity in its component Gibbs factors, and exponential decay of correlations and truncated expectations in the spread of distant families of random variables. In translation-invariant situations we obtain a central limit theorem. Our main tool is a series expansion in truncated expectations, which we analyze with L p methods.

X1,⋯,X> nare independent, identically distributed random variables with common density function f(x¦θ1,⋯,θk,θk+1), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that θk+1=b against the alternative that θk+1=b+cn−1/2. θ1,⋯,θkare nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦θ1-ai¦≦n−1/2M(n) for i=1,⋯,k, where a1, ⋯,akare known, and M(n) approaches infinity as n increases but n−1/2M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to θk+1=b, and weight 1-A to θk+1=b+cn−1/2, in each case the weight being spread uniformly over the possible values of θ1,⋯,θkin the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of θ1,...,θkare substituted for a1..., akrespectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a1..., akare.

In this paper are considered some conditions on locally compact topological groups that have arisen from the statistical analysis of group-invariant experiments and from probability theory. A catalogue is compiled of these conditions and of similar conditions that arise in the area of invariant means on locally compact topological groups. Known facts about the implication diagram of the conditions are described and some new implications are established. Counter-examples are discussed. An attempt is then made to throw some new light on the statistical relevance of the conditions in test theory, information theory and decision theory.

By direct probabilistic argument one term of an Edgeworth type asymptotic expansion is obtained for certain first passage distributions for random walks. These results provide partial justification for and extensions of approximations suggested earlier as a heuristic consequence of Laplace transform calculations.

The main results of the paper concern integral representations of convex sets of probability laws. Various types of simplices are introduced and characterized. Invariance with respect to Markovian operators plays a key role.

A non-uniform estimate of the rate of convergence in the central limit theorem for m-dependent random fields is obtained extending the work of Maejima (1978) for m-dependent random variables.

Gaussian processes satisfying Osterwalder-Schrader positivity are studied. A representation of the (generalized) covariance function of an OS-positive process as the Laplace transform of an operator-valued probability measure is given. It is shown that every Gaussian OS-positive process has a unique Gaussian canonical Markov extension. An explicit application is made to the generalized free Euclidean fields.

Limit theorems with a non-Gaussian limiting distribution have been obtained, under appropriate conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence by a number of people. The normalization has typically been nα, with 1/2

We show the existence of a time evolution {Pt; t∈ℝ of a locally perturbed equilibrium state P of infinitely many particles in ℝ{suv}, v≧1, evolving under the action of the infinite Newtonian dynamics associated to the same smooth, finite range pair potential as the equilibrium state itself. Moreover, it is shown that {Pt; t∈ℝ solves the weak BBGKY hierarchy equations. The treatment of this problem will be done in the general setting of so called (Σρ) point processes developed in [11, 10] and [4] and will require the method of moments.

Given a sequence of probability measures on a compact connected group we give an estimate on the speed of convergence to 1 of the density function of their n-fold convolution. Our method is an adaptation of the characteristic function technique to this case.

For a class of stationary independent increments processes we find a necessary and sufficient integral test on a function 0 0, where R= range of X, Rt = range of X up to epoch t.

In this paper we characterize the, what we call, visible projection of a point process on an arbitrary space as a Radon-Nikodym-derivative in the same manner the dual previsible projection of a process on ℝ+ is defined by Dellacherie and Meyer [1]; this visible projection turns out to coincide with the conditional intensity as defined by Papangelou [3]; a neat behaviour is imposed to the point process by only one intuitively clear condition, which is proved to be equivalent to the classical smoothnessconditions (∑) and (∑*).

A convenient method for proving weak convergence of a sequence of non-Markovian processesxe(·) to a jump-diffusion process is proved. Basically, it is shown that the limit solves the martingale problem of Strook and Varadhan. The proofs are relatively simple, and the conditions apparently weaker than required by other current methods (in particular, for limit theorems for a sequence of ordinary differential equations with random right hand sides). In order to illustrate the relative ease of applicability in many cases, a simpler proof of a known result on averaging is given.

The law of the iterated logarithm for posterior distributions and related Bayes estimators is studied for models which are differentiable in quadratic mean at the true parameter.

For independent identically distributed bivariate random vectors (X 1, Y 1), (X 2, Y 2), ... and for large t the distribution of X 1 +...+ X N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y 1, Y 2,.... Similar expansions are derived for multivariate X 1. Furthermore, local asymptotic expansions are valid for the distribution of f(X 1)+ ...+ f(X N ) when N is large and nonrandom, and X i , i=1, 2,..., is a discrete strongly mixing Markov chain.

Let G be the formal generator of a d-dimensional stochastic gradient system associated to an interaction potential U. If ∫ G ϕ d μ= 0 for such smooth ϕ that G ϕ IL1(μ), then certain moment conditions imply that μ is a Gibbs random field for U. If U satisfies a stability condition, and d≦2 or μ is translation invariant, then these moment conditions can be replaced by a natural support condition. Results of Holley and Stroock [6] are extended to certain unbounded spin systems.

Let {Yi} be iid with EY1=0, EY 1 2 =1. Let {Xi} be iid normal mean zero, variance one random variables. According to Strassen's first almost sure invariance principle {Xi} and {Yi} can be reconstructed on a new probability space without changing the distribution of each sequence such that \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = o(n{\text{ }}\log _2 n)^{1/2} } \) a.s., thus improving on the trivial bound obtainable from the law of the iterated logarithm: \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = O(n{\text{ }}\log _2 n)^{1/2} } \) a.s. In this work we establish analogous improvements for symmetric {Yi} in the domain of normal attraction to a symmetric stable law with index 0 γ≧α that \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = o(n^{1/\gamma } )} \) a.s., thus improving on the trivial bound: \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = o(n^{1/\alpha + \varepsilon } )} \) a.s., ɛ>0.

Distances between measures on IR d are determined from distances between their 1-dimensional projections. The method employed involves considering the 1-dimensional projections to be the Radon transform of the measures. Crucial to the main theorem is a continuity result for the inverse Radon transform. Focus is restricted to the Prohorov, dual bounded Lipschitz and d k metrics which metrize weak convergence of probability measures. These metrics are related to each other and to the Sobolev norms. The d k results extend from measures to generalized functions.

We introduce a martingale problem to associate diffusion processes with a kind of nonlinear parabolic equation. Then we show the existence and uniqueness theorems for solutions to the martingale problem.

The almost sure behaviour of the sample functions of the vectorvalued N-parameter Wiener process and its stable analogues is investigated, especially their small fluctuations. In particular, laws of the simple and of the iterated logarithm are established for the supremum of the local time increments or sojourn times. These results give precise information about the minimum oscillation of the sample functions. In addition, the Hausdorff measure problem for the graph and the range of these processes is solved.

A class of self-similar stationary random fields in ℝ d , d≧1 with finite variance is constructed by means of multiple stochastic integrals with respect to the Poisson random measure in ℝ d+1. Various topics associated with these fields such as subordination, ergodicity, existence of higher order moments, uniqueness of stochastic integral representation, renormalized powers of linear generalized fields and some limit theorems are studied. A Levy-Hincin type formula for the characteristic functional of general infinitely divisible self-similar random fields with finite variance is obtained.

Conditions have been given [7] under which a special automorphism over an automorphism admitting a simple approximation again admits a simple approximation, and so has simple spectrum. In this paper using different techniques to those employed in [7], we obtain improved results in the same direction. Specifically, conditions are given for a special automorphism over an automorphism, which admits either a simple approximation, or an approximation with suitable speed, to have bounded spectral multiplicity. Furthermore, we obtain as a corollary a result on primitive automorphisms, which partially generalises a result appearing in [4].

Almost sure limit functions of the properly normalised process, constructed out of the maximum of a sequence of Gaussian variables are obtained.

A Berry-Essen result and asymptotic expansions are derived for the distribution of bivariate von Mises functionals under moment and smoothness conditions. The results apply to the Cramer-von Mises ω 2 — statistic as well as to the Central Limit Theorem in Hilbert space, yielding a convergence rate O(n −1+ɛ) for every ɛ>0 on centered ellipsoids.

A class of non-pathological martingales is exhibited to which current martingale central limit theory does not apply. The problem consists in the convergence of the normalized sums of squares being only in distribution. The class is that of degenerate U-statistics.

For a strictly stationary random sequence (Xi)i≧0 we find sufficient conditions such that the distribution of the last exit time tβ = max{i∶ Xi>βi} (β>0) tends weakly to a nondegenerate limit distribution as β↓0.

A system of random variables(spins) Sx,x∈ℤ v, taking on values in ℝ is considered. Conditional probabilities for the joint distributions of a finite number of spins are prescribed; a DLR measure is then a process on the random variables which is consistent with the assigned conditional probabilities [1,2]. A case of physical interest both in Statistical Mechanics and in the lattice approximation to Quantum Field Theory is considered for which the spins interact pairwise via a potential JxySxSy, Jxy∈ℝ and via a self-interaction F(Sx), which, as ¦Sx¦→∞, diverges at least quadratically [3].

Extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction: (a) a weak “mixing condition” restricting long range dependence (b) a local condition restricting the “clustering” of high level exceedances.

Let E′ be the dual of a nuclear Frechet space E (of which the Schwartz space ℒ(ℝd) of tempered distributions is one). We give a simple sufficient condition for the strong continuity of a weakly continuous E′-valued stochastic process and as applications examine the Kolmogorov-Hahn-Delporte criteria.

Finite nearest particle systems are certain continuous time Markov chains on the collection of finite subsets ofZ1. In this paper, we give a sufficient condition for such a system to survive, in the sense that the probability of absorption at 0 is less than one. This theorem generalizes earlier results for the one-dimensional contact process.

Discrete parameter stationary processes with joint Gaussian distributions are considered. Loss is measured by squared error. It is shown that when estimating the value of g(x 0) of a spectral density function g the risk has an asymptotic lower bound similar to those previously obtained by Farrell [6] for the problem of density function estimation. A similar result is obtained for estimates of an entire function when risk is measured by integrated square error.

A characterization theorem based on the proportional relation between two truncated moments is proved for both continuous and discrete distributions. The results are applied for characterizing distributions of Pearson's system and its discrete analogon.

It is shown that for ϕ-mixing arrays of Banach space valued random vectors the central limit theorem implies the invariance principle. Applying this result to lattices of random variables a higher dimensional invariance principle under dependence assumptions is obtained.