Random discretization of stationary continuous time processes
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Random discretization of stationary continuous time processes Anne Philippe1
· Caroline Robet1 · Marie-Claude Viano2
Received: 12 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Gaussian process · Long memory · Partial sum · Random sampling · Regularly varying covariance
1 Introduction Most of the papers on time series analysis assume that observations are equally spaced in time. However, irregularly spaced time series data appear in many applications, for instance in astrophysics, climatology, high frequency finance, signal processing. Elorrieta et al. (2019) and Eyheramendy et al. (2018) propose generalisations of autoregressive models for irregular time series motivated by an application in astronomy. The spectral analysis of these data is also studied in many papers with applications into astrophysics, climatology, physics [see for instance Scargle (1982), Broersen (2007), Mayo (1978), Masry and Lui (1975)]. A possible way to address the problem of non-equally spaced data is to transform the data into equally spaced observations using some methods of interpolation [see for instance Adorf (1995), Friedman (1962), Nieto-Barajas and Sinha (2014)].
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Anne Philippe [email protected]
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Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
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Laboratoire Paul Painlevé UMR CNRS 8524, UFR de Mathematiques – Bat M2, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France
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An alternative one consists in assuming that time series can be embedded into a continuous time process. The data are then interpreted as a realization of a continuous temporal process observed at random times [see for instance Jones (1981), Jones and Tryon (1987), Brockwell et al. (2007)]. This approach requires the study of the effects of random sampling on the properties of continuous time process, as well as the development of inference methods for these models. Mykland (2003) studies the effects of the random sampling on the parameter estimation of time-homogeneous diffusion [see also Duffie and Glynn (2004)]. Masry (1994) establishes the properties of spectral density function estimation. In this paper we focus particularly on time series with long-range dependence [see Beran et al. (2013), Giraitis et al. (2012) for a review of available results for longmemory processes]. Some models and estimation methods have been proposed for continuous-time processes [see Tsai and Chan (2005a), Viano
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