\begin{theorem}[Chebyshev's inequality] % TODO Proof
    Let $X$ be a r.v.~with  $\Var(x) < \infty$.
    Then $\forall \epsilon > 0 : \bP \left[ \left| X - \bE[X] \right| > \epsilon\right] \le \frac{\Var(x)}{\epsilon^2}$.

\end{theorem}


We used Chebyshev's inequality. Linearity of $\bE$, $\Var(cX) = c^2\Var(X)$ and $\Var(X_1 +\ldots + X_n) = \Var(X_1) + \ldots + \Var(X_n)$ for independent $X_i$.


How do we prove that something happens almost surely?
\begin{lemma}[Borel-Cantelli]
    If we have a sequence of events  $(A_n)_{n \ge 1}$
    such that $\sum_{n \ge 1}  \bP(A_n) < \infty$,
    then $\bP[ A_n \text{for infinitely many $n$}] = 0$
    (more precisely: $\bP[\limsup_{n \to  \infty} A_n] = 0$).

    The converse also holds for independent events $A_n$.
    
\end{lemma}


Modes of covergence: $L^p$, in probability, a.s.