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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.

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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
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1 Answer

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.

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Piau: It seems you have misdefined $N$ above. –  user16344 Sep 19 '11 at 13:34
    
@djwayne ?? Can you be more specific? –  Did Jul 9 '12 at 16:55
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