# Sequence of characteristic functions

Let $\eta_k(t)$ be the characteristic function of a random variable $X_k$, for $k=1,2,...$ Consider a sequence of positive real numbers $c_1,c_2,...$ Take a function $g(t)=\sum\limits_{k=1} ^\infty c_k \eta_k(t)$. What are the necessary and sufficient conditions on the sequence ${c_k}$ s.t. $g$ is a characteristic function?

What I have in mind is setting a new function $$g_n(t):=\frac{\sum\limits_{k=1}^n c_k \eta_k(t)}{\sum\limits_{k=1}^n c_k} ,$$ then show that $g_n$ is a characteristic function, $g_n\rightarrow g$ pointwise, and $g$ is continuous at $0$.

It seems like your $g_n$ is unnecessary, monkey. I can tell you the condition: sum of $c_k$ is 1. – user16344 Sep 19 '11 at 13:34
Assume that the $c_k$ are positive and consider the condition (1) that $\sum\limits_kc_k=1$. If (1) holds, $g$ is the characteristic function of the random variable $X_N$, where $N$ is independent from the sequence $(X_k)$ and $\mathrm P(N=k)=c_k$ for every $k$. The other way round, assume $g$ is a characteristic function and note that $g(0)=\sum\limits_kc_k\eta_k(0)=\sum\limits_kc_k$ to deduce that (1) holds.
Thus, (1) is a necessary and sufficient condition for $g$ to be a characteristic function.
Piau: It seems you have misdefined $N$ above. – user16344 Sep 19 '11 at 13:34