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.
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