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
