Nonnormal deterministic equivalents and a transformation in stochastic mathematical programming

This paper considers random variables of the continuous type in a stochastic programming problem and presents (1) a general approach to the development of deterministic equivalents of constraints to be satisfied within certain probability limits, and (2) a deterministic transformation of a stochastic programming problem with random variables in the objective function. Deterministic equivalents are developed for constraints containing uniform random variables, but the approach used can be applied to other types of continuous random variables, as well. When the random variables appear in the objective function, a deterministic transformation of the stochastic programming problem is obtained to yield a closed-form solution without resort to a Monte Carlo computer simulation. Extension of this approach to stochastic problems with discrete random variables and integer decision variables is discussed briefly. A numerical example is presented.     

Nonsynchronous covariation process and limit theorems

An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.     

On χ2-test of goodness-of-fit for a class of discrete multivariate models

A model for an n-dimensional random vector which is the sum of v i.i.d. random variables with an exponential family distribution is considered. It is shown that the exact test can be approximated by a χ²-test as v → ∞. As an example a goodness-of-fit test for Poisson-distribution based on observed frequencies is derived.     

Self-intersection Local Time of Planar Brownian Motion Based on a Strong Approximation by Random Walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka?CRosen?CYor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.     

Joint Bayesian Analysis of Factor Scores and Structural Parameters in the Factor Analysis Model

A Bayesian approach is developed to assess the factor analysis model. Joint Bayesian estimates of the factor scores and the structural parameters in the covariance structure are obtained simultaneously. The basic idea is to treat the latent factor scores as missing data and augment them with the observed data in generating a sequence of random observations from the posterior distributions by the Gibbs sampler. Then, the Bayesian estimates are taken as the sample means of these random observations. Expressions for implementing the algorithm are derived and some statistical properties of the estimates are presented. Some aspects of the algorithm are illustrated by a real example and the performance of the Bayesian procedure is studied using simulation.     

Invariant functionals for random matrices

A new approach to the study of the Lyapunov exponents of random matrices is presented. It is proved that, under general assumptions, any family of nonnegative matrices possesses a continuous concave positively homogeneous invariant functional (“antinorm”) on ℝ₊^d. Moreover, the coefficient corresponding to an invariant antinorm equals the largest Lyapunov exponent. All conditions imposed on the matrices are shown to be essential. As a corollary, a sharp estimate for the asymptotics of the mathematical expectation for logarithms of norms of matrix products and of their spectral radii is derived. New upper and lower bounds for Lyapunov exponents are obtained. This leads to an algorithm for computing Lyapunov exponents. The proofs of the main results are outlined.     

Simulation of vector semibinary homogeneous random fields and modeling of broken clouds

A vector-valued homogeneous random field is said to be semibinary if its single-point marginal distribution is a sum of a singular distribution and a continuous one. In this paper, we present methods of numerical simulation of semibinary fields on the basis of the correlation structure and the marginal distribution. As an example we construct a combined model of cloud top height and optical thickness using satellite observations.     

随机半范模上等度连续的典则模同态族的特征

给出了随机半范模上等度连续的典则模同态族的一个待征——随机半范模上强凸的等度连续的典则模同态族可以被一个性质较好的连续随机半范数所控制．     

Strong deviation theorems for functionals of the nonnegative continuous random variables

By using the approach of Laplace transform, we establish the strong deviation theorems represented by inequalities and described by asymptotic average log-likelihood ratio for functionals of the nonnegative continuous random variables. As corollaries, we obtain several strong laws of large numbers for the random variable functionals. The results of this paper extend and improve the corresponding results of some current literatures.     

Expectation transfer between branching processes and random trees

An approach for translating results on expected parameter values from subcritical Galton–Watson branching processes to simply generated random trees under the uniform model is outlined. As an auxiliary technique for asymptotic evaluations, we use Flajolet's and Odlyzko's transfer theorems. Some classical results on random trees are re-derived by the mentioned approach, and some new results are presented. For example, the asymptotic behavior of linearly recursive tree parameters is described and the asymptotic probability of level k to contain exactly one node is determined. © 1993 John Wiley & Sons, Inc.     

Two-sex Branching Models with Random Control on the Number of Progenitor Couples

In this paper we introduce a class of two-sex branching models where, in each generation, a random control on the number of progenitor couples in the population is considered. For such a class, several probabilistic results are established. Also assuming offspring probability distribution belonging to the bivariate power series family, Bayes estimators for the mean vector and the covariance matrix of the offspring distribution are proposed. A computational method to determine highest posterior density credibility sets is stated. As illustration, a simulated example is provided.     

Local influence analysis of multivariate probit latent variable models

The multivariate probit model is very useful for analyzing correlated multivariate dichotomous data. Recently, this model has been generalized with a confirmatory factor analysis structure for accommodating more general covariance structure, and it is called the MPCFA model. The main purpose of this paper is to consider local influence analysis, which is a well-recognized important step of data analysis beyond the maximum likelihood estimation, of the MPCFA model. As the observed-data likelihood associated with the MPCFA model is intractable, the famous Cook's approach cannot be applied to achieve local influence measures. Hence, the local influence measures are developed via Zhu and Lee's [Local influence for incomplete data model, J. Roy. Statist. Soc. Ser. B 63 (2001) 111-126.] approach that is closely related to the EM algorithm. The diagnostic measures are derived from the conformal normal curvature of an appropriate function. The building blocks are computed via a sufficiently large random sample of the latent response strengths and latent variables that are generated by the Gibbs sampler. Some useful perturbation schemes are discussed. Results that are obtained from analyses of an artificial example and a real example are presented to illustrate the newly developed methodology.     

Coherent random permutations with biased record statistics

We consider random permutations that are defined coherently for all values of n, and for each n have a probability distribution which is conditionally uniform given the set of upper and lower record values. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family may be seen as an interpolation between two versions of Ewens’ distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random compositions.     

TTRISK: Tensor train decomposition algorithm for risk averse optimization

This article develops a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large-scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID-19. The results indicate that the risk-averse framework is feasible with the tensor approximations under tens of random variables.     

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

Various random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed-form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi-likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.     

Discretisation and duality of optimal Skorokhod embedding problems

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem.The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (2017), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.     

模糊随机需求的连续盘点存储策略研究

考虑到需求的模糊随机性,建立模糊随机需求情况下连续盘点存储策略的模糊随机成本模型。利用模糊随机变量的期望值理论,推导出了其成本期望值模型的解析表达式,进而给出了最优再订货点所属区间的判别条件以及最优再订货点和经济订货量的计算式;基于此,设计了一模糊随机需求的连续盘点最优存储策略算法。最后结合数值算例,分析了模糊随机需求概率分布及缺货成本对最优存储策略的影响。     

Singular Strictly Monotone Functions

We describe a universal approach to constructing continuous strictly monotone increasing singular functions on the closed interval [-1,1]. The “generator” of the method is the series ∑_k=1∞±2^-k with random permutation of signs, and the corresponding functions are generated as distribution functions of such series. As examples, we consider two stochastic methods of arranging signs: independent and Markov.     

Riesz-Nágy singular functions revisited

In 1952 F. Riesz and Sz.-Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-Nágy family of functions and we relate it to a system for real number representation which we call (τ,τ−1)-expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.     

Minimal cocycles with the scaling property and substitutions

‘Fractal’ functions are formulated as a minimal cocycle on a topological dynamics which admits nontrivial scaling transformations. In this paper, it is proved that if in addition it admits a continuous family of scaling transformations, then itscapacity is not ino(N ²). We define minimal cocycles with nontrivial scaling transformations coming from substitutions on a finite alphabet which are proved to have capacityO(N), so that they admit only a discrete family of scaling transformations. We also construct one which has capacityO(N ²) and admit a continuous family of scaling transformations. Supported by C.N.R.S.(U.R.A 225); Partially supported by the Japan Society for the Promotion of Science.