Random sampling of long-memory stationary processe

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

Random discretization of stationary continuous time processes

This paper investigates the second order properties of a stationarycontinuous time process after random sampling. While a short memory process gives alwaysrise to a short memory one, we prove that long-memory can disappearwhen the sampling law has very heavy tails. Despite the fact thatthe normality of the process is not maintained by random sampling, thenormalized partial sum process converges to the fractional Brownianmotion, at least when the long memory parameter is perserved

Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory

International audienceWe prove a functional central limit theorem for the empirical process of a stationary process $X_t = Y_t + V_t$, where $Y_t$ is a long memory moving average in i.i.d. r.v.'s $\zeta_s, s\le t$, and $V_t = V(\zeta_t, \zeta_{t-1}, \dots )$ is a weakly dependent nonlinear Bernoulli shift. Conditions of weak dependence of $V_t$ are written in terms of $L^2-$norms of shift-cut differences $V(\zeta_t, \dots, \zeta_{t-n}, 0, \dots, ) - V(\zeta_t, \dots, \zeta_{t-n+1}, 0, \dots )$. Examples of Bernoulli shifts are discussed. The limit empirical process is a degenerated process of the form $f(x) Z$, where $f$ is the marginal p.d.f. of $X_0$ and $Z$ is a standard normal r.v. The proof is based on a uniform reduction principle for the empirical process

Long memory properties and covariance structure of the EGARCH model

The EGARCH model of Nelson [29] is one of the most successful ARCH models which may exhibit characteristic asymmetries of financial time series, as well as long memory. The paper studies the covariance structure and dependence properties of the EGARCH and some related stochastic volatility models. We show that the large time behavior of the covariance of powers of the (observed) ARCH process is determined by the behavior of the covariance of the (linear) log-volatility process; in particular, a hyperbolic decay of the later covariance implies a similar hyperbolic decay of the former covariances. We show, in this case, that normalized partial sums of powers of the observed process tend to fractional Brownian motion. The paper also obtains a (functional) CLT for the corresponding partial sums' processes of the EGARCH model with short and moderate memory. These results are applied to study asymptotic behavior of tests for long memory using the R/S statistic