*Estimating a mean can be an arduous task. Observe, for instance, the “sudden jumps” in trajectories of the Monte Carlo estimate of , where .*

**Proposition.**

* Let be a sequence of i.i.d. and integrable random variables, and let . For all we have that
*

is or according to or .

**Corollary.**

*If , then converges towards at a rate almost surely slower than :*

*Proof of the proposition. *

We can suppose that , by the scale invariance of the problem. Note also that

from which we find

since almost surely. From the Borel-Cantelli lemma and

we obtain that in this case.

When , we let and note that . Therefore

Since the sum over of the right hand side probabilities converges (again using the fact that ), the Borel-Cantelli lemma yields the result.

**Remark.**

The result can be refined knowing that for some small : implies . The almost sure lower bound on the rate of convergence can be correspondingly adjusted.

Many thanks to Xi’an who wondered about the occurence of these jumps.

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