Let be $X$ an arbitrary random variable with characteristic function $\varphi(t)$. Prove that $$\psi(t)=\frac{1}{2-\varphi(t)},$$ and $$\chi(t)=\int_{0}^{\infty}\varphi(st)e^{-s}ds$$ are also characteristic functions.

The later is easy I think. It can be seen that it is continous at zero, $\chi(0)=1$ and positive definiteness can be derived. For any $n\in\mathbb N, \{u_i\}\in\mathbb R$ and $\{c_j\}\in\mathbb C$ where $i,j=0...n$ $$\sum_{i,j=1}^{n}c_i\bar{c_j}\int_{0}^{\infty}\varphi(su_i-su_j)e^{-s}ds=\int_{0}^{\infty}\sum_{i,j=1}^{n}c_i\bar{c_j}\varphi(su_i-su_j)e^{-s}ds\ge0$$ since the later is sum of non-negative numbers for any fixed $s$, given that $\varphi(t)$ is a charactheristic function, thus positive definite.

What is the trick with the first one?

  • 5
    $\begingroup$ $\frac 1 {2-x}$ is probability generating function of $G\sim Geom(\frac 1 2)$, hence $\psi(t)$ is characteristic function of $\sum_{i=1}^{G}X_i$ where $X$'s characteristic function is $\varphi(t)$. This can be obviously generalized to other compositions of functions. $\endgroup$
    – A.S.
    Mar 13, 2016 at 13:56
  • $\begingroup$ @A.S. Very interesting answer. I did not know this property. Do you have references ? $\endgroup$
    – Jean Marie
    Mar 13, 2016 at 14:01
  • 2
    $\begingroup$ @Jean $E\exp(it \sum_{i=1}^N X_i)=\sum_{k=0}^\infty P(N=k)\varphi_X^k(t)=pgf_N(\varphi_X(t))$. I've seen it used for compound Poisson process and Galton-Watson process. $\endgroup$
    – A.S.
    Mar 13, 2016 at 14:09


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