# Random Variable with Characteristic function $\frac{1}{2-\phi(t)}$

I am given that $$X$$ has c.f. $$\phi(t)$$, I need to find the random variable whose c.f. is equal to $$\frac{1}{2-\phi(t)}$$ in terms of $$X$$.

My idea is that express $$\frac{1}{2-\phi(t)}$$ as a series, since $$|\phi(t)| \leq 1$$ so we have $$\frac{1}{2-\phi(t)} = \sum_{n = 0}^{\infty}\frac{\phi(t)^n}{2^{n+1}}$$ From the question asked here, I am guessing that this c.f. corresponds to the random variable (I may be wrong):

$$Z = \sum_{n = 0}^{\infty}I(A = n)Z_n$$ where $$P(A = n) = \frac{1}{2^{n+1}}, \ n = 0,1,2,...$$ and $$Z_n = \sum_{i = 1}^{n}Y_i$$, $$Z_0 = 0$$, $$Y_i$$ are iid r.v.'s s.t. $$Y_i \sim X$$.

But I don't know how to prove this, can anyone point out a general direction? Thanks so much!

## 1 Answer

You already got the answer, you are just not completely writing that out. In probability context we usually write

$$\displaystyle Z = \sum_{i=1}^A Y_i$$

as a random sum, with the convention that $$Z = 0$$ when $$A = 0$$. We assumed $$A$$ is independent of $$Y_i$$ also, and they are defined as what you written. Then by law of total expectation, the characteristic function of $$Z$$ is

\begin{align} \phi_Z(t) &= E\left[\exp\left\{itZ\right\}\right] \\ &= E\left[\exp\left\{it\sum_{i=1}^A Y_i\right\}\right] \\ &= \sum_{n=0}^{\infty} E\left[\exp\left\{it\sum_{i=1}^A Y_i\right\} \Bigg| A=n \right] \Pr\{A = n\} \\ &= \Pr\{A = 0\} + \sum_{n=1}^{\infty} E\left[\exp\left\{it\sum_{i=1}^n Y_i\right\}\right] \Pr\{A = n\} \\ &= \frac {1} {2} + \sum_{n=1}^{\infty} E\left[\prod_{i=1}^n\exp\left\{it Y_i\right\}\right] \frac {1} {2^{n+1}} \\ &= \frac {1} {2} + \sum_{n=1}^{\infty} \prod_{i=1}^n E\left[\exp\left\{it Y_i\right\}\right] \frac {1} {2^{n+1}} \\ &= \frac {1} {2} + \sum_{n=1}^{\infty} \phi_X(t)^n \frac {1} {2^{n+1}} \\ &= \sum_{n=0}^{\infty} \frac {\phi_X(t)^n} {2^{n+1}} \\ &= \frac {1/2} {1 - \phi_X(t)/2} \\ &= \frac {1} {2 - \phi_X(t)} \end{align}

where line $$1$$ using the definition of characteristic function, line $$2$$ using the definition of $$Z$$, line $$3$$ using the law of total expectation, line $$4$$ using the independence of $$A$$ and $$Y_i$$ and the convention of the random sum, line $$5$$ using the basic property of exponential function, line $$6$$ using the independence of $$Y_i$$, line $$7$$ using the definition of characteristic function of $$Y_i$$ and they are identically distributed with the identical CF $$\phi_X$$, and the remaining lines are just some algebra to simplify the expression.

I have not study those complex analysis so if there is any holes in the above steps in regarding complex number, please help to fill them. I guess this should work.