A class of spectral density estimators

A class of spectral windows depending on one parameter is presented and shown to include many of the common windows. The mean square rate of convergence of the associated spectral density estimators are calculated in terms of this parameter for spectral densities which are locally Lipschitz continuous The class is shown to include certain data tapers and data windows corresponding to missing observations. This is true also for the kernels of (C−α) summability which provide means for estimating the spectral density when the covariance function is periodic.     

A canonical setting and separating times for continuous local martingales

The notion of a separating time for a pair of measures on a filtered space is helpful for studying problems of (local) absolute continuity and singularity of measures. In this paper, we describe a certain canonical setting for continuous local martingales (abbreviated below as CLMs) and find an explicit form of separating times for CLMs in this setting.     

A normal inverse Gaussian model for a risky asset with dependence

We present a new construction of the normal inverse Gaussian (NIG) fractal activity time model for a risky asset. The construction uses superpositions of diffusion processes and allows for specified exact NIG marginal distributions of the returns and flexible and tractable dependence structure including short or long range dependence. In the case of finite superposition, the fractal activity time is asymptotically self-similar, which is a desired feature seen in practice. The support for the distributional and dependence features of the risky asset model is provided by the data of currency exchange rates.     

Perfect simulation for locally continuous chains of infinite order

We establish sufficient conditions for perfect simulation of chains of infinite order on a countable alphabet. The new assumption, localized continuity, is formalized with the help of the notion of context trees, and includes the traditional continuous case, probabilistic context trees and discontinuous kernels. Since our assumptions are more refined than uniform continuity, our algorithms perfectly simulate continuous chains faster than the existing algorithms of the literature. We provide several illustrative examples.     

Conditional large deviations for a sequence of words

Cut an i.i.d. sequence (_Xi)$\left(X_i\right)$ of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional   empirical process of words, where one conditions on a typical underlying (_Xi)$\left(X_i\right)$. We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted.     

Fixed-gain estimation in continuous time

We prove that the parameter estimation error of continuous-time linear stochastic systems that is obtained in connection with a fixed-gain estimation method can be written as a stochastic integral plus a residual term, the moments of which are of order+o(1) where is the forgetting factor.     

Integrated mean square properties of density estimation by orthogonal series methods for dependent variables

Summary The rates at which integrated mean square and mean squre errors of nonparametric density estimation by orthogonal series method for sequences of strictly stationary strong mixing random variables are obtained. These rates are better than those known to hold for the independent case and they are shown to hold for Markov processes. In fact our results when specialized to the independent case are improvements over previously known results of Schwartz (1967,Ann. Math. Statist.,38, 1262–1265). An extension of the results to estimation of the bivariate density is also given. Research supported by a faculty summer research grant MS-STAT-42 from the University of Petroleum and Minerals.     

Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

In this paper, we study the non-parametric estimation of the invariant density of some ergodic hamiltonian systems, using kernel estimators. The main result is a central limit theorem for such estimators under partial observation (only the positions are observed). The main tools are mixing estimates and refined covariance inequalities, the main difficulty being the strong degeneracy of such processes. This is the first paper of a series of at least two, devoted to the estimation of the characteristics of such processes: invariant density, drift term, volatility.     

A note on the de La Vallée Poussin criterion for uniform integrability

In this paper, we present new versions of the classical de La Vallée Poussin criterion for uniform integrability. Our results concern the uniform integrability of a continuous function relative to a sequence of distribution functions. We apply our results to obtain a result on the convergence of a sequence of integrals which we illustrate with an example.     

Strong consistency of density estimation by orthogonal series methods for dependent variables with applications

Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables. The results obtained are then applied to two estimates of the functional Δ(f)=∫f ² (x)dx were strong consistency is established. One of the suggested two estimates of Δ(f) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where they established consistency in the second mean of the estimate. Research supported in part by the National Research Council of Canada and in part by McMaster University Research Board. Now at Memphis State University, Memphis, Tennessee 38152, U.S.A.     

The composition of Wiener functionals with non absolutely continuous Shifts

Summary In this paper we study some classes of Wiener functionals whose elements can be composed with a non-linear, non-absolutely continous transformation of the form of perturbation of identity in the direction of Cameron-Martin space. We show that under certain conditions the image of the Wiener measure under the above transformation induces a generalized Wiener functional on certain Sobolev spaces generalizing the Radon-Nikodym relation to non absolutely continuous transformations. A series representation for the generalized Radon-Nikodym derivative is presented and conditional expectations of some generalized random variables are considered.     

Recursive equations for the predictive distributions of some determinantal processes

This paper provides recursive equations for the predictive distributions of one-dependent and two-dependent determinantal processes. Fixed order recursive equations can be applied both to efficiently simulate trajectories and to explore properties of the process.     

A note on using periodogram-based distances for comparing spectral densities

Motivated by a recent paper of Caiado et al. (2009), we investigate testing problems for spectral densities of time series with unequal sample sizes. We thereby focus on analyzing their mathematical properties and illustrate our results in a small simulation study.     

A convergence theorem for spectral factorization

This paper presents a convergence theorem for an iterative method of spectral factorization in the context of multivariate prediction theory. It may be viewed as a constructive proof that the factorization exists, using only the analytic results of Hardy space theory.     

A continuous spectral density for a random field of continuous-index

Linear dependence coefficients are defined for random fields of continuous-index, which are modified from those already defined for random fields indexed by an integer lattice. When a selection of these linear dependence conditions are satisfied, the random field will have a continuous spectral density function. Showing this involves the construction of a special class of random fields using a standard Poisson process and the original random field.     

A multistate monotone system signature

Using decomposition’s results for a multistate monotone system (MMS) we get a MMS signature at a system level. As in the classical case and through exchangeability properties we obtain the Samaniego signature and representations theorems for system reliability.     

Asymptotic results on a class of adaptive multi-treatment designs

The play-the-winner (PW) rule is an important method in clinical trials where patients can be assigned to one of the two treatments. In the PW rule, the probability of the next patient to be assigned to a particular treatment only depends on the response of the current patient. In this paper, we consider a general kind of PW rule for multi-treatment adaptive designs, in which the probability that a treatment is assigned to the next patient depends upon both the response of the previous patient and an estimated parameter, e.g., the observed success rate. Using this kind of adaptive designs, more information of previous stages are used to update the model at each stage, and more patients may be assigned to better treatments. The strong consistency and the asymptotic normality are established for the allocation proportions.     

Measure concentration for a class of random processes

Summary.   Let X={X _i } _{i =−∞} ^∞ be a stationary random process with a countable alphabet and distribution q. Let q ^∞(·|x _{− k} ⁰) denote the conditional distribution of X ^∞=(X ₁,X ₂,…,X _n ,…) given the k-length past:

Large deviations for lattice systems I. Parametrized independent fields

Summary We prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter. This dependence respects the action of the shift group, and a large deviation principle holds whenever a certain ergodicity condition is satisfied. We also investigate the entropy functions for these processes, especially in relation to the usual relative entropy.