In an attempt to derive something, I have come to a point where I need to know what the following can be simplified to. It looks to me that I am looking for the 4th moment of ''y''. I am just not sure how to go about this.
$$E\space[\space(\space\sum_{i=1}^N (\space y_i^2\space)\space)^2\space]= \space ?$$
For my problem, the following relationship is valid$$ E\space[\sum_{i=1}^N\space y_i^2 \space] = \space N\sigma^2 $$
Note that: $$y_i \sim \space N(0,\sigma^2) $$
It's not a fourth moment it's actually a second (non-central) moment of the chi square.
Up to a constant the sum $\sum(y_i^2)$ is a chi-squared random variable with $N$ degress of freedom. So up to a constant this is the second non-central moment of a chi square with $N$ degrees of freedom.
Look here for the formula for the second non-central moment: https://en.wikipedia.org/wiki/Chi-squared_distribution#Noncentral_moments