On the embeddability of certain infinitely divisible probability measures on Lie groups

We describe certain sufficient conditions for an infinitely divisible probability measure on a Lie group to be embeddable in a continuous one-parameter semigroup of probability measures. A major class of Lie groups involved in the analysis consists of central extensions of almost algebraic groups by compactly generated abelian groups without vector part. This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain connected Lie groups, including the so called Walnut group. The embeddability is concluded also under certain other conditions. Our methods are based on a detailed study of actions of certain nilpotent groups on special spaces of probability measures and on Fourier analysis along the fibering of the extension.     

Measurements and infinite-dimensional statistical inverse theory

The most important ingredient of the statistical inverse theory is the indirect and noisy measurement of the unknown. Without the measurement, the formula for the posterior distribution becomes useless. However, inserting the measurement into the posterior distribution is not always simple. In the general setting, the posterior distribution is defined as a regular conditional probability. Hence it is known up to almost all measurements, which is inconvenient when we are given a single measurement. This shortage is covered in the finite-dimensional statistical inverse theory by fixing versions of probability density functions. A usual choice is to consider continuous probability density functions. Unfortunately, infinite-dimensional probability measures lack density functions which prohibits us from using the same method in the general setting. In this work, other possibilities for fixing the posterior distributions are discussed in the Gaussian framework. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse

Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.     

On solutions of minimax test problems for special capacities

Summary Solutions to minimax test problems between neighbourhoods generated by specially defined capacities are discussed. The capacities are superpositions of probability measures and concave functions, so the paper covers most of the earlier results of Huber and Rieder concerning minimax testing between -contamination and total variation neighbourhoods. It is shown that the Neyman-Pearson lemma for 2-alternating capacities, proved by Huber and Strassen, can be applied to test problems between noncompact neighbourhoods of probability measures. It turns out that the Radon-Nikodym derivative between the special capacities is usually a nondecreasing function of the truncated likelihood ratio of some probability measures.     

A Differential Free Point Generation Scheme in the Differential Evolution Algorithm

In this paper we derive the probability distribution of trial points in the differential evolution (de) algorithm, in particular the probability distribution of points generated by mutation. We propose a point generation scheme that uses an approximation to this distribution. The scheme can dispense with the differential vector used in the mutation of de. We propose a de algorithm that replaces the differential based mutation scheme with a probability distribution based point generation scheme. We also propose a de algorithm that uses a probabilistic combination of the point generation by the probability distribution and the point generation by mutation. A numerical study is carried out using a set of 50 test problems, many of which are inspired by practical applications. Numerical results suggest that the new algorithms are superior to the original version both in terms of the number of function evaluations and cpu times.     

Singularity among selfsimilar Gaussian random fields with different scaling parameters and others

Abstract

It is shown in this paper that the probability measures generated by selfsimilar Gaussian random fields are mutually singular, whenever they have different scaling parameters. So are those generated from a selfsimilar Gaussian random field and a stationary Gaussian random field. Certain conditions are also given for the singularity of the probability measures generated from two Gaussian random fields whose covariance functions are Schoenberg–Lévy kernels, and for those from stationary Gaussian random fields with spectral densities.     

Renewal-theoretic considerations in evaluation of forecast monitoring schemes

To evaluate the performance of a scheme for monitoring forecasts that are generated by exponential smoothing, forecasters have traditionally used a simulation-based estimator of some characteristic of the associated run-length distribution. The most frequently cited performance measures are the average run length and the probability that the run length does not exceed a user-specified cut-off point. However, there is disagreement about the definition of run length that is appropriate in the context of forecasting. In this paper we use fundamental results from renewal theory to establish the precise relationships between conflicting formulations both of the average run length and of the probability density function for the run length. More generally we derive the asymptotic mean and the asymptotic distribution function of the forward recurrence time for both ordinary and delayed renewal processes whose interoccurrence distributions are arithmetic with a given span. We discuss the practical significance of these results in the context of forecasting. Our findings bear directly on the way in which simulation experiments should be designed and executed to compare alternative forecast monitoring schemes.     

关于从闭区间到完备随机赋范模的抽象值函数的Riemann 可积性的进一步研究

郭铁信和张霞最近引入和研究了从一个闭区间到一个完备随机赋范模的抽象值函数的Riemann积分, 证明了值域几乎处处有界的连续函数是Riemann 可积的. 本文首先给出该结果的一个更简短的证明, 使得我们对于值域的几乎处处有界性有一个更深的认识, 受此启发, 我们进一步构造两个例子, 其一说明值域并非几乎处处有界的连续函数也可以是Riemann 可积的, 另一例子说明连续函数可以非Riemann 可积. 最后, 我们证明从一闭区间到一个满支撑的完备随机赋范模的所有连续函数都Riemann 可积的充要条件是基底概率空间本质上由至多可数原子生成.     

一阶格点系统的不变测度与Liouville型方程

该文讨论一阶格点系统的解在相空间中的概率分布问题.作者先证明该格点系统的解算子生成的过程存在拉回吸引子,然后证明拉回吸引子上存在唯一的Borel不变概率测度,且该不变测度满足Liouville型方程.     

The Hausdorff Moment Problem under Finite Additivity

We investigate to what extent finitely additive probability measures on the unit interval are determined by their moment sequence. We do this by studying the lower envelope of all finitely additive probability measures with a given moment sequence. Our investigation leads to several elegant expressions for this lower envelope, and it allows us to conclude that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions.     

A modified HOL priority scheduling discipline: Performance analysis

In this paper, we introduce and analyze a modified HOL (head-of-the-line) priority scheduling discipline. The modification is incorporated to cope with the so-called starvation problem of regular HOL priority queues. We consider a discrete-time single-server queueing system with two priority queues of infinite capacity and with the introduced priority scheme. We show that the use of probability generating functions is suitable for analyzing the system contents and the packet delay. Some performance measures (such as means and variances) of these stochastic quantities will be derived. Furthermore, approximate expressions of the tail probabilities are obtained from the probability generating functions, by means of the dominant-singularity method. These expressions, together with their characteristics, constitute one of the main contributions of this paper. Finally, the impact and significance of the m-HOL (modified HOL) priority scheduling on these performance measures is illustrated by some numerical examples.     

LARGE DEVIATIONS FOR STATIONARY φ-MIXING SEQUENCES IN τ-TOPOLOGY

The results of Bryc on large deviations for empirical measures of stationary Φ-mixing sequences are extended. Bryc states his results in the usual weak topology on the space ofprobability measures. In this paper, under somewhat weaker assumptions than those of Bryc,the author extends Bryc's results by taking the finer topology which is generated by the integralsover bounded measurable functions.     

Robust probability updating

This paper discusses an alternative to conditioning that may be used when the probability distribution is not fully specified. It does not require any assumptions (such as CAR: coarsening at random) on the unknown distribution. The well-known Monty Hall problem is the simplest scenario where neither naive conditioning nor the CAR assumption suffice to determine an updated probability distribution. This paper thus addresses a generalization of that problem to arbitrary distributions on finite outcome spaces, arbitrary sets of ‘messages’, and (almost) arbitrary loss functions, and provides existence and characterization theorems for robust probability updating strategies. We find that for logarithmic loss, optimality is characterized by an elegant condition, which we call RCAR (reverse coarsening at random). Under certain conditions, the same condition also characterizes optimality for a much larger class of loss functions, and we obtain an objective and general answer to how one should update probabilities in the light of new information.     

Convergent Cutting-Plane and Partial-Sampling Algorithm for Multistage Stochastic Linear Programs with Recourse

We propose an algorithm for multistage stochastic linear programs with recourse where random quantities in different stages are independent. The algorithm approximates successively expected recourse functions by building up valid cutting planes to support these functions from below. In each iteration, for the expected recourse function in each stage, one cutting plane is generated using the dual extreme points of the next-stage problem that have been found so far. We prove that the algorithm is convergent with probability one.     

Bisexual Galton‐Watson Branching Process in Varying Environments

Abstract

In this paper we introduce a bisexual Galton‐Watson branching process (BGWP) in which the offspring probability distribution is different in each generation. We obtain some relations among the probability generating functions (pgf) involved in the model and, making use of mean growth rates and fractional linear functions (flf), we provide sufficient and necessary conditions for its almost sure extinction.     

A discrete-time queueing system with optional LCFS discipline

We consider a discrete-time queueing system in which the arriving customers decide with a certain probability to be served under a LCFS-PR discipline and with complementary probability to join the queue. The arrivals are assumed to be geometrical and the service times are arbitrarily distributed. The service times of the expelled customers are independent of their previous ones. We carry out an extensive analysis of the system developing recursive formulae and generating functions for the steady-state distribution of the number of customers in the system and obtaining also recursive formulae and generating functions for the stationary distribution of the busy period and sojourn time as well as some performance measures.     

Functions operating on multivariate distribution and survival functions—With applications to classical mean-values and to copulas

Functions operating on multivariate distribution and survival functions are characterized, based on a theorem of Morillas, for which a new proof is presented. These results are applied to determine those classical mean values on [0,1]ⁿ which are distribution functions of probability measures on [0,1]ⁿ. As it turns out, the arithmetic mean plays a universal rôle for the characterization of distribution as well as survival functions. Another consequence is a far reaching generalization of Kimberling’s theorem, tightly connected to Archimedean copulas.     

Genetic algorithms in statistical tolerancing

An important aspect in manufacturing design is the distribution of geometrical tolerances so that an assembly functions with given probability, while minimising the manufacturing cost. This requires a complex search over a multidimensional domain, much of which leads to infeasible solutions and which can have many local minima. As well, Monte-Carlo methods are often required to determine the probability that the assembly functions as designed. This paper describes a genetic algorithm for carrying out this search and successfully applies it to two specific mechanical designs, enabling comparisons of a new statistical tolerancing design method with existing methods.     

A stable stieltjes technique for computing orthogonal polynomials and jacobi matrices associated with a class of singular measures

A recursive technique for the determination of Jacobi matrices associated with multifractal measures generated by Iterated Function Systems is described. This technique allows for the stable determination of large-rank matrices, a task for which the conventional approach, classical polynomial sampling, is proven here to be severely ill-conditioned. Application to the integration of smooth functions is presented in a physical example, and relevance of this new technique to the study of the asymptotic properties of orthogonal polynomials of singular measures is discussed. Further applications to the study of almost periodic quantum systems are briefly mentioned.     

Generalized binomial and negative binomial distributions of order<Emphasis Type="Italic">k</Emphasis> by the<Emphasis Type="Italic">l</Emphasis>-overlapping enumeration scheme

In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number of trials and derive the probability generating functions. The present work extends several properties of distributions of orderk and leads us a new type of geneses of the discrete distributions.