Sudden jumps and almost surely slow convergence

Estimating a mean can be an arduous task. Observe, for instance, the “sudden jumps” in trajectories of the Monte Carlo estimate \overline X_n of \mathbb{E}X, where \mathbb{E}X^2 = \infty.

sudden jumpsProposition.
Let X_1, X_2, X_3, \dots be a sequence of i.i.d. and integrable random variables, and let \overline X_k = \frac{1}{k}\sum_{i=1}^k X_i. For all M > 0 we have that

P\left( \left| \overline X_{n} - \overline X_{n-1}\right| > Mn^{-1/2} \;\; \text{ i.o.} \right)

is 1 or 0 according to \text{Var}(X_n) =\infty or \text{Var}(X_n) < \infty.

Corollary.
If \text{Var}(X_k) = \infty, then \overline X_n converges towards \mathbb{E}X_k at a rate almost surely slower than \frac{1}{\sqrt{n}} :

\left|\overline X_n - \mathbb{E}X_n \right| \not = \mathcal{O}\left(n^{-1/2}\right) \quad \text{a.s.}

PDF proof.

 

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

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