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 $0$.
Thank you for your help.