Let $\phi:\mathbb{R^2} \rightarrow \mathbb{C}$ be a measurable function such that $\forall y \in \mathbb{R}: \phi(.,y):x \mapsto \phi(x,y)$ is a characteristic function. Let $f$ be a density function on $\mathbb{R}.$
Prove that $h(x)=\int_{\mathbb{R}}\phi(x,y)f(y)\,\mathrm dy$ is a characteristic function.
I am trying to find a random variable $X$ such that $\varphi_X(x)=h(x)$.
If $Y$ is a random variable with density $f,$ then $h(x)=\int_{\mathbb{R}}\phi(x,y)\,\mathrm dP_Y(y)$ but this is not helpful.
So is there a random variable $X$ such that $\varphi_X(x)=h(x)?$
I don't think you can find explicitly a random variable with characteristic function $h$. You can prove that $h$ is a characteristic function using Bochner's Theorem. It is trivial to check that $h$ is a positive definite function so we only have to show that $h(x) \to 1$ s $x \to 0$. But this is easy from DCT: $\phi(x,y) \to 1$ as $ x\to 0$ for each $y$ and $|\phi(x,y) | \leq 1$ for all $x,y$.