Distribution sum of product of normal distribution and bernoulli distribution:

Question

I wanted to understand the distribution of the following summation --

$$\sum_{i=1}^n a_i x_i z_i^{(k)}.$$ where $a_i \in R$; $x_i \sim\mathcal{N}(0, 1)$, that is, $x_i$ is sampled from a normal distribution with unit variance and zero mean, and $z_i^{(k)}$ is an indicator random variable that takes value $1$ with probability $\frac{1}{k}$, and $0$ otherwise.

Also, does that distribution of the summation is close to the normal distribution for small values of $k$?

Answer 1

These things are generally quite difficult. If $1/k$ is small, and $n \approx k$, then what you have is close to the product of a Poisson random variable and an independent Gaussian random variable. And for the correct choice of parameters, this is well approximated by the product of two Gaussian random variables. But the product of two independent Gaussian random variables is not Gaussian.