whether the following $$\varphi (t) = -\cos^{4} t+2\cos^{2}t$$ is a characteristic function (short for ch.f.) of some random variable $X$ ?
Soppose $\varphi (t)$ is a ch.f. of a random variable $X$,
then $$\frac{\mathrm{d} }{\mathrm{d} x}\varphi (t)|_{t=0}=i\mathbb{E}X=0;\quad\frac{\mathrm{d^2} }{\mathrm{d} x^2}\varphi (t)|_{t=0}=i^2\mathbb{E}X^2=0.$$ Thus $X$ has some symmetric distribution and $\int _{\mathbf{R}}X^2(\omega)\mathbb{P}(d\omega)=0.$ $$\int _{\mathbf{R}}X^2(\omega)\mathbb{P}(d\omega)=0\Longrightarrow \mathbb{P}(X=0)=1\Longrightarrow \text{The ch.f of }X:\phi_{X}(t)=1\ne\varphi (t).$$
I am not sure the above solution is right. I need some help to verifty it.