Notation: $S_n = X_1 + \ldots + X_n$
Definition $1$ : The sequence of random variables $X_1, X_2, \ldots$ satisfies the Weak Law of Large Numbers if $$\frac{S_n – E[S_n]}{n} \overset{p}{\to} 0$$
Definition $2$: The sequence of random variables $X_{1}, X_{2}, \ldots$ satisfies the Strong Law of Large Numbers if $$\frac{S_n – E[S_n]}{n} \overset{\text{a.s.}}{\to} 0$$
Exercise: Let $(X_{n})$ be a sequence of independent random variables with $P(X_n = n^2) = \frac{1}{n^3}$, $P(X_n = 0) = 1 – \frac{1}{n^3}$. Does this sequence satisfy the Weak Law of Large Numbers? Does it satisfy the Strong Law of Large Numbers?
I've tried to find the distribution for $S_n$, but this approach didn't work. Any suggestions/hints on how to do this exercise are really appreciated.
Best Answer
Note that $E(S_n)=\sum_{j=1}^nEX_j=\sum_{j=1}^n\frac 1j$ behaves as $\log n$, hence $\frac{E(S_n)}n$ converges to $0$.
Let $E_k:=\{X_k=k^2\}$. Since $\sum_j\frac 1{j^3}$ is convergent, by Borel-Cantelli's lemma, $P(\limsup_j E_j)=0$. This means that there is $\Omega'\subset \Omega$ of probability $1$ such that for each $\omega\in\Omega'$, there is $N(\omega)$ for which $X_k(\omega)\neq k^2$, hence $X_k(\omega)=0$ whenver $k\geqslant N(\omega)$. This means that $\frac{S_n(\omega)}n=\frac 1n\sum_{j=1}^{N(\omega)}X_j(\omega)\to 0$.