Continuous-time Stationary Processes Research Articles

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

On the Value of Information and Mean Squared Error for Noisy Gaussian Models

The relationship between the age of information (AoI) and the mean squared error (MSE) in optimisation problems has been widely investigated for various Gaussian Markov models. Recently, we put the AoI in an information-theoretic context and proposed a mutual information-based value of information (VoI) framework to characterise how valuable the status updates are. In this letter we consider general noisy Gaussian models and investigate the connection between the VoI and MSE. We show that the VoI can be presented by a logarithmic function of the MSE. We also study the rate of change of the two with respect to the signal-to-noise ratio (SNR) and the AoI. The results hold for both discrete-time and continuous-time stationary Gaussian processes. This letter illuminates a useful relationship between information theory and signal processing theory in the presence of the AoI and additive noise, and provides a possible way to infer the information value by estimating the MSE for practical applications.

Asymptotic distribution of the score test for detecting marks in hawkes processes

The score test is a computationally efficient method for determining whether marks have a significant impact on the intensity of a Hawkes process. This paper provides theoretical justification for use of this test. It is shown that the score statistic has an asymptotic chi-squared distribution under the null hypothesis that marks do not impact the intensity process. For local power, the asymptotic distribution against local alternatives is proved to be non-central chi-squared. A stationary marked Hawkes process is constructed using a thinning method when the marks are observations on a continuous time stationary process and the joint likelihood of event times and marks is developed for this case, substantially extending existing results which only cover independent and identically distributed marks. These asymptotic chi-squared distributions required for the size and local power of the score test extend existing asymptotic results for likelihood estimates of the unmarked Hawkes process model under mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for the exponential decay Hawkes process.

Modeling and Estimation of Multivariate Discrete and Continuous Time Stationary Processes

In this paper, we give an autoregressive model of order 1 type of characterization covering all multivariate strictly stationary processes indexed by the set of integers. Consequently, under square integrability, we derive continuous time algebraic Riccati equations for the parameter matrix of the characterization. This provides us with a natural way to define the corresponding estimator. In addition, we show that the estimator inherits consistency from autocovariances of the stationary process. Furthermore, the limiting distribution is given by a linear function of the limiting distribution of the autocovariances. We also present the corresponding existing results of the continuous time setting paralleling them to the discrete case treated in this paper.

Random discretization of stationary continuous time processes

This paper investigates second order properties of a stationary continuous time process after random sampling. While a short memory process always gives rise to a short memory one, we prove that long-memory can disappear when the sampling law has very heavy tails. Despite the fact that the normality of the process is not maintained by random sampling, the normalized partial sum process converges to the fractional Brownian motion, at least when the long memory parameter is preserved.

Estimating the conditional expectations for continuous time stationary processes

Estimating the conditional expectations for continuous time stationary processes

Goodness-of-fit Tests for Continuous-time Stationary Processes

The paper considers the following problem of hypotheses testing: based on a finite realization {X(t)}, 0 ≤ t ≤ T of a zero mean real-valued mean square continuous stationary Gaussian process X(t), t ϵ R, construct goodness-of-fit tests for testing a hypothesis H0 that the hypothetical spectral density of the process X(t) has the specified form. We show that in the case where the hypothetical spectral density of X(t) does not depend on unknown parameters (the hypothesis H0 is simple), then the suggested test statistic has a chi-square distribution. In the case where the hypothesis H0 is composite, that is, the hypothetical spectral density of X(t) depends on an unknown p–dimensional vector parameter, we choose an appropriate estimator for unknown parameter and describe the limiting distribution of the test statistic, which is similar to that of obtained by Chernov and Lehman in the case of independent observations. The testing procedure works both for short- and long-memory models.

Embedding in law of discrete time ARMA processes in continuous time stationary processes

Given any stationary time series {Xn:n∈Z} satisfying an ARMA(p,q) model for arbitrary p and q with infinitely divisible innovations, we construct a continuous time stationary process {xt:t∈R} such that the distribution of {xn:n∈Z}, the process sampled at discrete time, coincides with the distribution of {Xn}. In particular the autocovariance function of {xt} interpolates that of {Xn}.

On the robustness to small trends of parameter estimation for continuous-time stationary models with memory

The paper deals with a question of robustness of inferences, carried out on a continuoustime stationary process contaminated by a small trend, to this departure from stationarity.We show that a smooth periodogram approach to parameter estimation is highly robust to the presence of a small trend in themodel. The obtained result is a continuous version of that of Hede and Dai (Journal of Time Series Analysis, 17, 141-150, 1996) for discrete time processes.

Asymptotic variance of stationary reversible and normal Markov processes

We obtain necessary and sufficient conditions for the regularvariation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in n.

The trace problem for Toeplitz matrices and operators and its impact in probability

The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szegö, Toeplitz forms and their applications (University of California Press, Berkeley, 1958). It has then been extensively studied in the literature. In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes. The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, hypotheses testing about the spectrum, etc. We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory.

Information, Estimation, and Lookahead in the Gaussian Channel

We consider mean squared estimation with lookahead of a continuous-time signal corrupted by additive white Gaussian noise. We show that the mutual information rate function, i.e., the mutual information rate as function of the signal-to-noise ratio (SNR), does not, in general, determine the minimum mean squared error (MMSE) with fixed finite lookahead, in contrast to the special cases with 0 and infinite lookahead (filtering and smoothing errors), respectively, which were previously established in the literature. We also establish a new expectation identity under a generalized observation model where the Gaussian channel has an SNR jump at $t=0$, capturing the tradeoff between lookahead and SNR. Further, we study the class of continuous-time stationary Gauss-Markov processes (Ornstein-Uhlenbeck processes) as channel inputs, and explicitly characterize the behavior of the minimum mean squared error (MMSE) with finite lookahead and signal-to-noise ratio (SNR). The MMSE with lookahead is shown to converge exponentially rapidly to the non-causal error, with the exponent being the reciprocal of the non-causal error. We extend our results to mixtures of Ornstein-Uhlenbeck processes, and use the insight gained to present lower and upper bounds on the MMSE with lookahead for a class of stationary Gaussian input processes, whose spectrum can be expressed as a mixture of Ornstein-Uhlenbeck spectra.

Consistency results for the kernel density estimate on continuous time stationary and dependent data

The aim of this paper is to study the consistency of the kernel density estimator pertaining to a continuous time stationary process X=(Xt)t≥0, with an underlying density f. More precisely, in a rather general dependency setting, where we use a martingale difference device and a technique based on a sequence of projections on σ-fields, we establish the almost sure pointwise and uniform consistencies with rates of the estimate fT of f built upon the part (Xt)0≤t≤T of the process X.

Trace Approximations of Products of Truncated Toeplitz Operators

The paper establishes error orders for integral limit approximations to the traces of products of truncated Toeplitz operators generated by integrable real symmetric functions defined on the real line. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous-time stationary processes (asymptotic distributions and large deviations of Toeplitz-type quadratic functionals, estimation of the spectral parameters and functionals, etc.). An explicit second-order asymptotic expansion is found for the trace of a product of two truncated Toeplitz operators generated by the spectral densities of continuous-time stationary fractional Riesz–Bessel motions. The order of magnitude of the second term in this expansion is shown to depend on the long-memory parameters of the processes. Also, it is shown that the pole in the first-order approximation is removed by the second-order term, which provides a substantially improved approximation to the original functional.

Piecewise linear density estimation for sampled data

Nonparametric density estimation is considered for a discretely observed stationary continuous-time process. For each of three given time sampling procedures either random or deterministic, we establish that histograms and frequency polygons can reach the same optimal L 2-rates as in the independent and identically distributed case. Moreover, thanks to a suitable “high frequency” sampling design, these rates are derived together with a minimized time of observation depending on the regularity of sample paths.

Improved estimator of the continuous-time kernel estimator

There have been many papers presenting kernel density estimators for a strictly stationary continuous time process observed over the time interval [0, T]. However the estimators do not satisfy the property of mean-square continuity if the process is mean-square continuous. In this paper we present a modified kernel estimator and substantiate that the modified estimator satisfies the property of mean-square continuity. In a simulation study the results show the modified estimator is better than the original estimator in some cases.

The Role of Information in Repeated Games With Frequent Actions

We show that in repeated interactions the avenues for effective provision of incentives depend crucially on the type of information players observe. We establish this conclusion for general repeated two-player games in which information arrives via a continuous-time stationary process that has a continuous multidimensional Brownian component and a Poisson component, and in which the players act frequently. The Poisson jumps can be used to effectively provide incentives both with transfers and value burning, while continuous Brownian information can be used to provide incentives only with transfers.

Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields

We show how to detect optimal Berry--Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein's method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75--118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395--406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551--579]); (ii) to ``exploding'' quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer--Major CLT for functionals of a fractional Brownian motion.

Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities

The paper considers the mean square linear prediction problem for some classes of continuous-time stationary Gaussian processes with spectral densities possessing singularities. Specifically, we are interested in estimating the rate of decrease to zero of the relative prediction error of a future value of the process using the finite past, compared with the whole past, provided that the underlying process is nondeterministic and is “close” to white noise. We obtain explicit expressions and asymptotic formulae for relative prediction error in the cases where the spectral density possess either zeros (the underlying model is an anti-persistent process), or poles (the model is a long memory processes). Our approach to the problem is based on the Krein’s theory of continual analogs of orthogonal polynomials and the continual analogs of Szego theorem on Toeplitz determinants. A key fact is that the relative prediction error can be represented explicitly by means of the so-called “parameter function” which is a continual analog of the Verblunsky coefficients (or reflection parameters) associated with orthogonal polynomials on the unit circle. To this end first we discuss some properties of Krein’s functions, state continual analogs of Szego “weak” theorem, and obtain formulae for the resolvents and Fredholm determinants of the corresponding Wiener-Hopf truncated operators.

Error bounds for approximations of traces of products of truncated Toeplitz operators

The paper establishes error orders for integral limit approximations to the traces of products of truncated Toeplitz operators generated by integrable real symmetric functions defined on the real line. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous-time stationary processes (asymptotic distributions and large deviations of Toeplitz type quadratic functionals, estimation of the spectrum, etc.). The results improve the rates obtained by the authors (in an earlier paper).