Is this use of a Law of Large Numbers correct? The usual LLN I've seen, the theorem states that $1/n \sum u_t \rightarrow^p E(u_t)=\mu$,where the expected value is independent of t. However, I cannot apply it to the exercise below. 

In the picture, the author seems to be using $ \displaystyle \frac{1}{n} \sum u_t \rightarrow^p \lim_{n \rightarrow \infty}\frac{1}{n} \sum E(u_t)$



Result (3.44) is $Var(\hat u_t^2)=E(\hat u_t^2)=(1-h_t)\sigma^2$, and it's also the result they refer to as 'in the previous exercise', and (3.46) is just the first equation, without the plims.

So, what LLN can I apply?

Any help would be appreciated.
Are the $\hat{u}_t^2$ independent? If yes, the result is indeed a version of the law of large numbers due to Kolmogorov, and that deals with independent (but not necessarily identically distributed) processes.

Here is the formal statement. Suppose that $\{u_t\}_{t=1}^{\infty}$ is a sequence of independent real-valued random variables that satisfies 
\begin{equation*}
\sum_{t=1}^{\infty}{\dfrac{Var(u_t)}{t^2}}&lt;\infty
\end{equation*}

Then the random variable
\begin{equation*}
\dfrac{1}{n} \sum_{t=1}^{n}{[u_t-\mathbb{E}(u_t)]}
\end{equation*}
converges to 0 almost surely.

Edit: the assumption of independence seems stronger than needed, and the following result (an adapted version of the law of large numbers for non-independent processes) might be useful. If $\sigma_{n,m}=Cov(u_n,u_m)$ exists, and if there is $0&lt; \alpha &lt;1$ and $M \in \mathbb{R}$ such that $\sigma_{n,m} &lt; M \alpha^{|n-m|}$ for all $(n,m)$, then
\begin{equation*}
\mathbb{P}(|\dfrac{1}{n}\sum_{t=1}^{n}u_t- \dfrac{1}{n}\sum_{t=1}^{n}\mathbb{E}(u_t)|&gt;\epsilon] &lt; \dfrac{M}{n \epsilon^2}
\end{equation*}
for any $\epsilon$.