Random sampling of long-memory stationary processes

Abstract

This paper investigates the second order properties of a stationary 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 heavy enough tails. We prove that under rather general conditions the existence of the spectral density is preserved by random sampling. We also investigate the effects of deterministic sampling on seasonal long-memory.

Inference for continuous-time long memory randomly sampled processes

“…We should mention that randomly spaced observations have been used by Bardet and Bertrand (2010) in estimating the spectral density of Gaussian processes with stationary increments. We also note that in light of previously cited work in discrete time, there was an effect of the random sampling on the dependence structure of the process, in particular Philippe and Viano (2010) showed that the intensity of the long memory is preserved when the distribution of sampling intervals has a finite moment, but there are also situations where a reduction of the long memory is observed. We conclude this introduction by mentioning that Philippe et al (2018) established a rather surprising characteristic consisting in the loss of the joint-Gaussienty of the sampled process when the original process was Gaussian.…”

Random discretization of stationary continuous time processes

“…In this paper, we study the properties of this. In particular, we show that the results obtained by Philippe and Viano (2010) on the auto-covariance function are preserved for continuous time process X. The large-sample statistical inference relies often on limit theorems of probability theory for partial sums.…”

“…The phenomenon is the same as in the discrete case (see Philippe and Viano (2010)): starting from a long memory process, a heavy tailed sampling distribution can lead to a short memory process.…”

“…In the light of previous results in discrete time, there was an effect of the random sampling on the dependence structure of the process. Indeed, Philippe and Viano (2010) show that the intensity of the long memory is preserved when the law of sampling intervals has finite first moment, but they also pointed out situations where a reduction of the long memory is observed.…”

Asymptotic behavior of functionals of cyclic long-range dependent random fields

“…Limit theorems for functionals of long-range dependent Gaussian processes were investigated by M. Rosenblatt [16], Dobrushin and Major [3,4], Taqqu [17,18], Giraitis and Surgailis [6,7], Oppenheim and Viano [5,14,15] and others. For multidimensional results of this type see [9,10,13,19].…”