Edit: I needed to revise this question as it was suggested.

Suppose there are $N$ realizations of Gaussian process denoted as the vectors $z_{j} \in \mathbb{R}^{n}$ for $j = 1, \ldots, N$. Let $y$ be a random variable such that $y = \sum_{j=1}^{N}(Bz_{j})[i]$ where $B$ is a unitary matrix. What is the variance of $y^{2}$?

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