I was reading about white noise and it stated:
Although $\varepsilon_t$ & $y_t$ are serially uncorrelated, they are not necessarily serially independent, because they are not necessarily normally distributed. If in addition to being serially uncorrelated, $y$ is serially independent, then we say $y$ is independent white noise.
I cannot understand the link between the normal distribution and being serially independent. Can anyone help me with this?
Best Answer
My interpretation of the (slightly paraphrased) statement that the OP is reading, viz.
is that the author is asserting (quite correctly) that uncorrelated random variables need not be independent but the reason for this failure of uncorrelatedness to imply independence is that we cannot assert that the uncorrelated random variables are normally distributed. If the author is not implying that uncorrelated random variables that are normally distributed are independent random variables, then he is certainly begging the reader to jump to that (false) conclusion, by musing in the the subjunctive mood: "Gol dang it, if only those pesky $Y_n$'s were normally distributed in addition to being uncorrelated, then we could take those uncorrelated (and normal) $Y_n$'s to be independent random variables and avoid a lot of headaches." However, Moderator @whuber has stated (in a comment on the main question) that he does not interpret that sentence that way, and that the statement quoted by the OP is perfectly accurate.
In my opinion, the second sentence quoted by the OP.
also incorrect. Independent random variables are always uncorrelated and it is unnecessary to start with uncorrelated random variables and then impose the additional constraint that they are independent random variables. Furthermore, if by serially independent it is meant that for all $n\neq m$, $Y_m$ and $Y_n$ are independent random variables (that is, only pairwise independence is required), then I disagree vehemently with the assertion that $\{Y_n\}$ is a white noise process. For a random process to be called a white noise process, the random variables need to be mutually independent, not just pairwise independent, and most people, upon encountering the phrase white noise process, are likely to assume that the the random variables constituting the white noise process also are zero-mean random variables with common finite variance $\sigma^2$. This property of the $Y_n$'s is nowhere mentioned in the paragraph fragment quoted by the OP.
Finally, turning to the OP's complaint
I say that it is a red herring. Uncorrelated (marginally) normal random variables are not necessarily independent random variables while uncorrelated jointly normal random random variables are always independent random variables.
A standard counterexample begins with $X \sim N(0,1)$ and an independent random variable $Z$ that takes on values $+1$ and $-1$ with equal probability. Then $Y = XZ$ is also a standard normal random variable since \begin{align}P\{Y \leq x\} &= P\{X \leq x\mid Z=+1\}P\{Z=+1\} + P\{X \geq -x\mid Z=-1\}P\{Z=-1\}\\ &= \frac 12 \Phi(x) + \frac 12 (1 - \Phi(-x))\\ &= \Phi(x). \end{align} Also, $\quad\operatorname{cov}(X,Y) = E[XY]-E[X]E[Y]= E[X^2Z]=E[X^2]E[Z]=0$ showing that $X$ and $Y$ are are uncorrelated random variables. But $X$ and $Y$, although they are uncorrelated normal random variables, are not independent random variables but instead very much dependent random variables since given that $X=x$, $Y$ takes on values $\pm x$ with equal probability $\frac 12$.
Now, with $\{Z_n\}$ being a sequence of independent random variables with the same distribution as $Z$ (and all independent of $X$ also), set $Y_n = XZ_n$. It follows from our construction that $Y_n \sim N(0,1)$. But, $$E[Y_nY_{m}] = E[X^2Z_nZ_m] = E[X^2]E[Z_n]E[Z_m] = 0~\text{provided that} ~ n \neq m.$$ Thus, the $\{Y_n\}$ are uncorrelated and normal. But they are not independent because if we know that $Y_m = y$, then we know that $Y_n$ is necessarily either $y$ or $-y$.
Normal random variables need not be jointly normal random variables and it is only in the case of joint normality that one can assert that uncorrelated (jointly) normal random variables are independent.