Calculating the Variance of the Product of Two Standard Normally Distributed Vectors

Question

I am trying to compute the variance of the inner product of two vectors, $V_1$ and $V_2$, both of size $m$ and with independent elements that follow a standard normal distribution [$v_i \sim \mathcal N(0, 1)$]. Any help would be appreciated.

Answer 1

$V_1 = [a_1, \ldots, a_m]$ and $V_2 = [b_1, \ldots, b_m]$

$ X = \langle V_1, V_2 \rangle = \sum_{i=1}^m a_i b_i$

$\mathbb{E}[X] = \sum_{i=1}^m \mathbb{E}[a_i b_i] = 0$ if the elements of $V_1$ and $V_2$ are independent, otherwise there is insufficient information to proceed.

$\mathbb{E}[X^2] = \sum_{i=1}^m \mathbb{E}[a_i^2 b_i^2] + \sum_{i=1}^m \sum_{j=1, j \neq i}^m \mathbb{E}[a_i b_j]$

Again, under the independence assumption, the second term vanishes and the first term becomes $m$. Cannot proceed without the independence assumption.

Update: If $a_i$ and $b_i$ are independent, $\mathbb{E}[a_i^2 b_i^2] = \mathbb{E}[a_i^2]\mathbb{E}[b_i^2]$

Since $a_i \sim N(0,1), \mathbb{E}[a_i^2] = 1$