If x,)...) X, are independent identically distributed Rd-valued random vectors with probability measure p and empirical probability measure p,, and if QZ is a subset of the Bore1 sets on Rd, then we show that P{sup,, ~

For stochastic programs with recourse and with (several joint) probabilistic constraints, respectively, we derive quantitative continuity properties of the relevant expectation functionals and constraint set mappings. This leads to qualitative and quantitative stability results for optimal values and optimal solutions with respect to perturbations of the underlying probability distributions. Earlier stability results for stochastic programs with recourse and for those with probabilistic constraints are refined and extended, respectively. Emphasis is placed on equipping sets of probability measures with metrics that one can handle in specific situations. To illustrate the general stability results we present possible consequences when estimating the original probability measure via empirical ones.

Abstract: In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We

In this paper, stochastic programming problems are viewed as parametric programs with respect to the probability distributions of the random coefficients. General results on quantitative stability in parametric optimization are used to study distribution sensitivity of stochastic programs. For recourse and chance constrained models quantitative continuity results for optimal values and optimal solution sets are proved (with respect to suitable metrics on the space of probability distributions). The results are useful to study the effect of approximations and of incomplete information in stochastic programming.

Using results from parametric optimization, we derive for chance-constrained stochastic programs quantitative stability properties for locally optimal values and sets of local minimizers when the underlying probability distribution is subjected to perturbations in a metric space of probability measures. Emphasis is placed on verifiable sufficient conditions for the constraint-set mapping to fulfill a Lipschitz property which is essential for the stability results. Both convex and nonconvex problems are investigated. For a chance-constrained model of power dispatch, where the power demand enters as a random vector with incompletely known probability distribution, we discuss consequences of our general results for the stability of optimal generation costs and optimal generation policies.

The purpose of these lectures is to present three different approaches with their own methods for establishing uniform laws of large numbers and uniform ergodic theorems for dynamical systems. The presentation follows the principle according to which the i.i.d. case is considered first in great detail, and then attempts are made to extend these results to the case of more general dependence structures. The lectures begin (Chapter 1) with a review and description of classic laws of large numbers and ergodic theorems, their connection and interplay, and their infinite dimensional extensions towards uniform theorems with applications to dynamical systems. The first approach (Chapter 2) is of metric entropy with bracketing which relies upon the Blum-DeHardt law of large numbers and Hoffmann-Jørgensen’s extension of it. The result extends to general dynamical systems using the uniform ergodic lemma (or Kingman’s subadditive ergodic theorem). In this context metric entropy and majorizing measure type conditions are also considered. The second approach (Chapter 3) is of Vapnik and Chervonenkis. It relies

Abstract-A test sequence is used to select the best rule from a rich class of discrimination rules defined in terms of the training sequence. The Vapnik-Chervonenkis and related inequalities are used to obtain distribution-free bounds on the difference between the probability of error o € the selected rule and the probability of error of the best rule in the given class. The bounds are used to prove the consistency and asymptotic optimality for several popular classes, including linear discriminators, nearest neighbor rules, kernel-based rules, histogram rules, binary tree classifiers, and Fourier series classifiers. In particular, the method can be used to choose the smoothing parameter in kernel-based rules, to choose k in the k-nearest neighbor rule, and to choose between parametric and nonparametric rules. Index Terms-Automatic parameter selection, empirical risk, error estimation, nonparametric rule, probability of error, statistical pattern recognition, Vapnik-Chervonenkis inequality.