Estimating a mean can be an arduous task. Observe, for instance, the “sudden jumps” in trajectories of the Monte Carlo estimate of , where .
Let be a sequence of i.i.d. and integrable random variables, and let . For all we have that
is or according to or .
If , then converges towards at a rate almost surely slower than :
Many thanks to Xi’an who wondered about the occurence of these jumps.