# 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 $\mathbb{E}X^2 = \infty$.

Proposition.
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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