Let $\phi_k(t)$ be the characteristic function of a random variable $X_k$, $k = 1,2,\dots$. Consider a set of positive real numbers $\{p_1, p_2, \dots \}$, take a function:
$$\phi(t) = \sum_{k=1}^{\infty}p_k\phi_k(t)$$
What is the conditions on $\{p_1, p_2, \dots \}$ such that $\phi$ is a characteristic function?
I know that a characteristic function need to satisfy following properties:
$$\phi(0) = 1$$
$$\phi(-t) = \overline{\phi(t)}$$
$$|\phi(t)|=\left|E[e^{itX}]\right|\leq E|e^{itX}|=1$$
$$|\phi(t+h)-\phi(t)|\leq E|e^{ihX}-1|$$
$$E[e^{it(aX+b)}]=e^{itb}\phi(at)$$
But I think only satisfy these properties will not ensure that $\phi(t)$ is a characteristic function for some random variables. What is the direction should I approach this problem?
Thank you very much for the help.
Best Answer
Considering that $\phi(0)=1$ and that $\phi_k(0)=1$ for every $k$, one sees that the condition $$\sum_kp_k=1$$ is necessary. To prove that is also sufficient, and to avoid most of the technicalities, consider some random variables $N$ and $(X_k)$ defined on the same probability space, $N$ independent of $(X_k)$, each $X_k$ with characteristic function $\phi_k$, and $N$ integer valued with distribution $P(N=k)=p_k$ for every $k$. Then the random variable $$X_N=\sum_kX_k\mathbf 1_{N=k},$$ has characteristic function $$\phi=\sum_kp_k\phi_k.$$ In particular, $\phi$ is a characteristic function.