I need to find out if $\phi$ is a characteristic function
$$ \phi(t_1,t_2) = (p\cos(t_1+t_2) + 1 - p)^n, \quad 0 < p < 1, n \in \mathbb{N_{>0}} $$
What I've tried:
- Instead I'm trying to prove that $p\cos(t_1+t_2) + 1 - p$ is characteristic because if it is then it's n-th power also is characteristic.
- I've tried converting $\cos$ to trig form and then doing Fourier transform to get p.d.f. and got $$ p \pi \delta_{x-1} \delta_{y-1} + p \pi \delta_{x+1} \delta_{y+1} + 2 \pi \delta_{x} \delta_{y} + 2 p \pi \delta_{x} \delta_{y} $$ but I'm not sure if this is a p.d.f.
- Also, $\phi$ looks like a characteristic function of $Bin(n,p)$: $$ {\displaystyle \!\,(1-p+pe^{it})^{n}} $$ but I don't know if there is any connection between them.
You have the right hunch.
You can check the characteristic function of the following random variable: $$X=(B_1+...+B_n,B_1+...B_n)$$ where $B,B_1,...,B_n$ are iid random variables, B equals 0 with probability 1-p, B equals 1 with probability p/2, -1 with probability p/2