Taking averages

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When I simulate a stochastic process say, 1000 times and the output has any distribution. I then take the sample mean of these 1000 outputs and do it again 1000 times. SO I end up with 1M data-points and 1000 averages. These 1000 averages happen to be normally distributed. I know there is a name for this occurence but I forgot. Do you guys know?