[Math] Is $e^{2(\cos(t)-1)}$ the characteristic function of some random variable

Question

I am asked to decide whether
$$f(t)=e^{2(\cos(t) -1)}$$
is the characteristic function of some random variable.

Attempt. I am trying to find directly a possible associated random variable (which might be a totally bad strategy). The assignment actually starts like this:

Let $X$, $Y$ be two i.i.d. random variables with characteristic function $\varphi_X$ ($=\varphi_Y$). Compute $\varphi_{-Y}$ and $\varphi_{X-Y}$ in terms of $\varphi_{X}$.

It's immediate to see that $\varphi_{X-Y}(t)=\varphi_X(t)\varphi_X(-t)$. Thus I can maybe use this observation to guess that, if $f(t)=\varphi_X(t)$ for some r.v. $X$, then $X=Y-Z$ where $\varphi_Y(t)=e^{\cos(t) -1}=e^{\cos(-t) -1}=\varphi_{-Z}(t)$.

Therefore it doesn't seem a bad idea to look for some density function $g_Y(x)$ such that
$$e^{\cos t -1} = \int_{\mathbb R} e^{itx} g_Y(x) dx = \int_{\mathbb R} \cos (tx) \ g_Y(x) dx + i \int_{\mathbb R} \sin (tx) \ g_Y(x) dx.$$
Since $e^{\cos t -1} \in \mathbb R$ for all $t \in \mathbb R$, we have that $i \int_{\mathbb R} \sin (tx) \ g_Y(x) dx$ must be $0$. Hence a good choice for $g_Y(x)$ could be an even function, defined on a symmetric domain (w.r.t. $0$).

Here I'm stuck. It seems to me that I am still quite far from a solution. "Even function on a symmetric domain" is clearly a quite small achievement (and it's even just a guess).

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Please note that we didn't discuss Bochner's theorem in class. We did discuss Inversion Theorem, but apparently computations get quite messy.

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I'd prefer to receive just hints for the moment, not full solutions. Thank you for your help.

Answer 1

Hints:

• If $Y$ is Bernoulli symmetric with $P(Y=1)=P(Y=-1)=\frac12$ then $\varphi_Y(t)=$ $____$.
• If $N$ is Poisson with parameter $\lambda$, then the generating function of $N$ is $$g_N(s)=E(s^N)=\sum_{n\geqslant0}s^n\mathrm e^{-\lambda}\frac{\lambda^n}{n!}=\mathrm e^{-\lambda}\sum_{n\geqslant0}\frac{(\lambda s)^n}{n!}=\text{____}.$$
• If $Y$ and $(Y_n)$ is i.i.d. and $N$ is Poisson with parameter $\lambda$ and independent of $(Y_n)$, then $$X=\sum\limits_{n=1}^NY_n$$ is such that, for every $n\geqslant0$, $$E(\mathrm e^{\mathrm itX}\mid N=n)=E(\mathrm e^{\mathrm it(Y_1+Y_2+\cdots+Y_n)})=E(\mathrm e^{\mathrm itY_1})^n=\varphi_Y(t)^n,$$ hence $$\varphi_X(t)=E(E(\mathrm e^{\mathrm itX}\mid N))=E(\varphi_Y(t)^n)=g_N(\varphi_Y(t)),$$
• Finally, $\varphi:t\mapsto\mathrm e^{-2(1-\cos(t))}$ is the characteristic function of $X=$ $____$.