# Is gaussian random process multiplied by a random cosine a gaussian random process

Let $${U(t)}$$ be a gaussian random process.

Let $$\phi$$ be a random variable such that $$\phi \sim \text{Uniform}(0,2\pi)$$.

$$U(t)$$ is independent of $$\phi$$.

Let $$Y(t)$$ be a random process that created by $$Y(t) = U\left(t\right)\cdot\cos\left(2\pi f_{c}t+\phi\right)$$

Is $$Y(t)$$ a gaussian random process?

I know that to show that a random process is gaussian I need to show that for every linear combination with times $$(t_1,t_2,\dots t_n)$$ the linear combination of : $$Q=\sum_{i=1}^{n}\alpha_{i}U\left(t_{i}\right)\cos\left(2\pi f_{c}t_{i}+\phi\right)$$ is a gaussian random variable.

But in this case the $$\phi$$ is random so how do I know if it's a gaussian random variable or not?

In general, a gaussian random variable $$X$$ with mean $$E[X] = 0$$ and $$Var(X)=\sigma^2$$ must satisfy: $$E[X^4]=3\sigma^4$$ In our case we can calculate for $$Q = U(0)\cos(2 \pi f_c \cdot 0 + \phi)=U(0)\cos(\phi)$$ Calculating $$E[Q^4]$$ does not satisfy the constraint.