Random discretization of stationary continuous time processes

Abstract

This paper investigates the second order properties of a stationary continuous time process after random sampling. While a short memory process gives always 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 perserved.

Parameter estimation for random sampled Regression Model with Long Memory Noise

“…We illustrate the convergence speed of the parameter estimations with graphical analysis. In Figure (1) we can see that for different values of H, the estimation of the parameter reaches in very few iterations.…”

“…Fractional Brownian motion B H with Hurst parameter H ∈ (1/2, 1) is a centered Gaussian process with covariance structure is given in (1). It is well-known that if H = 1/2, then B H is a standard Brownian motion.…”

“…We take in this case the value a = 2. Tables (1) and (2) present the simulation results for estimation in case 1: Jittered sampling (according to Equation 4)). Performance statistics presented are the mean, standard deviation (SD) and Kurtosis.…”

“…They proved a central limit theorem and provided an application to biological data. Philippe et al in [1] gave one of the last works on this topic, where the authors considered the study of the preservation of memory in a statistical model.…”

On the Consistency of Least Squares Estimator in Models Sampled at Random Times Driven by Long Memory Noise: The Jittered Case

“…They prove a central limit theorem providing an application to biological data. Philippe, A., et al (2020) give the latest works on this topic, the authors consider the study of the preservation of memory in a statistical model.…”