A thinning result for pure jump processes

A standard thinning procedure for point processes is extended to processes of pure jump type in which each jump is retained with probability p or deleted with probability 1 ? p, independently of everything else.Two theorems are proved, the first gives a sufficient condition for the existence of thinned pure jump processes, the second concerns the convergence of such processes to pure jump processes whose increments are generated by a Cox process. Some generalizations are discussed.     

Detection of a change point based on local-likelihood

In this paper, we consider the regression function or its νth derivative in generalized linear models which may have a change/discontinuity point at an unknown location. The location and its jump size are estimated with the local polynomial fits based on one-sided kernel weighted local-likelihood functions. Asymptotic distributions of the proposed estimators of location and jump size are established. The finite-sample performances of the proposed estimators with practical aspects are illustrated by simulated and beetle mortality examples.     

Quantile regression with ℓ 1—regularization and Gaussian kernels

The quantile regression problem is considered by learning schemes based on ? ₁—regularization and Gaussian kernels. The purpose of this paper is to present concentration estimates for the algorithms. Our analysis shows that the convergence behavior of ? ₁—quantile regression with Gaussian kernels is almost the same as that of the RKHS-based learning schemes. Furthermore, the previous analysis for kernel-based quantile regression usually requires that the output sample values are uniformly bounded, which excludes the common case with Gaussian noise. Our error analysis presented in this paper can give satisfactory convergence rates even for unbounded sampling processes. Besides, numerical experiments are given which support the theoretical results.     

Forward equations for reflected diffusions with jumps

In this paper we obtain the forward equations associated with the evolution of the density, if it exists, of reflected diffusions on the positive orthant with jumps which form a marked point process whose random jump measure possesses a stochastic intensity. These results generalize the so-called generalized Dynkin equations for piecewise deterministic jump processes due to Davis. We then consider the stationary case where the existence of a stochastic intensity is not needed. The techniques are based on local times and the use of random jump measures. We discuss the application of these results to problems arising in queuing and storage processes as well as stationary distributions of diffusions with delayed and jump reflections at the origin.This research was supported in part by the Quebec-France Cooperative Research Program and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP 0042024.     

On the Equivalence of Probability Spaces

For a general class of Gaussian processes W, indexed by a sigma-algebra \({\mathscr {F}}\) of a general measure space \((M,{\mathscr {F}}, \sigma )\), we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering \(\sigma (A)\), for \(A\in {\mathscr {F}}\), as a quadratic variation of W over A. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: (i) a computation of generalized Ito integrals and (ii) a proof of an explicit and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path space, and the other where it is Schwartz’ space of tempered distributions.     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

Fractional Brownian motion approximation based on fractional integration of a white noise

We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We consider correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similarity properties of the approximation to fractional Brownian motion, namely, `τ^H laws' for the structure function and the range. We conclude that the models proposed serve as a convenient tool for modelling of natural processes and testing and improvement of methods aimed at analysis and interpretation of experimental data.     

一般状态空间跳过程的强遍历性

研究了一般状态空间跳过程的强遍历性,利用最小非负解理论及马氏性,得到了强遍历性的几个等价条件,把连续时间可数状态马氏链的相关结果推广到一般状态空间跳过程的情形.     

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

A univariate Hawkes process is a simple point process that is self-exciting and has a clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes processes have wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.     

A Non‐Gaussian Ornstein–Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing

A mean‐reverting model is proposed for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non‐Gaussian Ornstein–Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. In a simulation example it is demonstrated that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. The pricing of European call and put options written on electricity forward contracts is also discussed.     

Uniform Laws of Large Numbers for Triangular Arrays of Function-indexed Processes under Random Entropy Conditions

Mean Glivenko Cantelli Theorems are established for triangular arrays of rowwise independent processes. Methods developed by Pollard (1990) are combined with a truncation method essentially due to Alexander (1987). By this, applicability to partial sum processes in particular is achieved, for which Pollard’s truncation method fails. Nevertheless, the metric entropy condition appearing here is kept as weak as Pollard’s by means of application of Hoffmann-Jørgensen’s inequality, which has not been used so far in this context. The main theorem of the paper contains Pollard’s theorem as well as former results by Giné and Zinn (1984) and proves applicable to so-called random measure processes, certain function-indexed processes including empirical processes, partial-sum processes, the sequential empirical process and certain types of smoothed empirical processes. Statistical applications include nonparametric regression and the estimation of the intensity measure of a spatial Poisson process (Poisson point process).     

Existence of the linear prediction for Banach space valued Gaussian processes

The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.     

Limit set of inhomogeneous Ornstein-Uhlenbeck processes,destabilization and annealing

We consider the Diffusion Process obtained by perturbing a dynamical system having a single equilibrium point x, by a fixed time-inhomogeneous Gaussian process whose intensity tends to 0 at infinity. We establish criteria for the exit time from a neighborhood of x to be a.s. finite by linking this fact with the structure of the limit set at infinity. We are also able to compute this limit set for inhomogeneous Ornstein-Uhlenbeck processes associated to linear systems. An application is given to simulated annealing.     

Local Asymptotic Normality Property for Ornstein–Uhlenbeck Processes with Jumps Under Discrete Sampling

We address the issue of the local asymptotic normality property and the Fisher information for three characterizing parameters of Ornstein–Uhlenbeck processes with jumps under low frequency and high frequency discrete sampling with expanding observation window. The martingale method with the Kolmogorov backward equation and the Malliavin calculus are employed to derive explicit formulas for derivatives of the likelihood ratio function in the form of conditional expectation, which serve as essential tools for justifying the passage to the limit by the dominated convergence theorem. This approach makes it possible to carry out the proof without specifying the law of the jump component and without knowing the tail behaviors of the transition probability density and, as a consequence, to keep various types of jump structure within the scope of this article. The Fisher information under high-frequency sampling is essentially identical to the one for purely Gaussian Ornstein–Uhlenbeck processes due to the dominance of the Gaussian component over the jump component in the short time framework.     

Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case

Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation. For one direction of the equivalence an explicit Markov kernel is constructed.     

一般状态空间跳过程的Harris常返

研究了一般状态空间跳过程的Harris常返,利用马氏性,得到了跳过程Harris常返的几个等价条件.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

Integrating DEA-oriented performance assessment and target setting using interactive MOLP methods

Data envelopment analysis (DEA) and multiple objective linear programming (MOLP) are tools that can be used in management control and planning. Whilst these two types of model are similar in structure, DEA is directed to assessing past performances as part of management control function and MOLP to planning future performance targets. This paper is devoted to investigating equivalence models and interactive tradeoff analysis procedures in MOLP, such that DEA-oriented performance assessment and target setting can be integrated in a way that the decision makers’ preferences can be taken into account in an interactive fashion. Three equivalence models are investigated between the output-oriented dual DEA model and the minimax reference point formulations, namely the super-ideal point model, the ideal point model and the shortest distance model. These models can be used to support efficiency analysis in the same way as the conventional DEA model does and also support tradeoff analysis for setting target values by individuals or groups. A case study is conducted to illustrate how DEA-oriented efficiency analysis can be conducted using the MOLP methods and how such performance assessment can be integrated into an interactive procedure for setting realistic target values.     

A universal envelope for Gaussian processes and their kernels

The central theme in our paper deals with mathematical issues involved in the answer to the following question: How can we generate stochastic processes from their correlation data? Since Gaussian processes are determined by moment information up to order two, we focus on the Gaussian case. Two functional analytic tools are used here, in more than one variant. They are: operator factorization; and direct integral decompositions in the form of Karhunen-Loève expansions. We define and study a new interplay between the theory of positive definition functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. The three classes of processes we study are as follows: Processes associated with: (a) arbitrarily given sigma finite regular measures on a fixed Borel measure space; (b) with Hilbert spaces of sigma-functions; and (c) with systems of self-similar measures arising in the theory of iterated function systems. Starting with a non-degenerate positive definite function K on some fixed set S, we show that there is a choice of a universal sample space Ω, which can be realized as a “boundary” of (S,K). Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.     

Even Unimodular Gaussian Lattices of Rank 12

We classify even unimodular Gaussian lattices of rank 12, that is, even unimodular integral lattices of rank 12 over the ring of Gaussian integers. This is equivalent to the classification of the automorphisms τ with τ²=−1 in the automorphism groups of all the Niemeier lattices, which are even unimodular (real) integral lattices of rank 24. There are 28 even unimodular Gaussian lattices of rank 12 up to equivalence.