Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.