In probability theory, the characteristic function of a random variable $X$ is defined as
$$
\varphi_X(t) =\mathrm{E}\left(e^{\mathrm{i}tX}\right),
$$
We further know that the characteristic function of the sum of $N$ independent random variables, $Z=\sum_{k=1}^{N}X_k$, is simply
$$
\varphi_Z(t)=\mathrm{E}\left(e^{itZ}\right)=\prod_{k=1}^N\phi_{X_k}(t).
$$
Thus, the product of characteristic functions has a clear interpretation in terms of the transformation of the sum of independent random variables.
For quite some time now, I am wondering whether the sum of characteristic functions may also have an interpretation in terms of random variables and their probability distributions?
For example, we know that the sum of an exponential and a standard normal random variable has the transform
$$
\varphi_{Exp(\lambda)+N(\mu,\sigma)}(t)=\frac{1}{1-i\lambda t}e^{i\mu t – \frac{1}{2}t^2\sigma^2}
$$
But is there a meaningful interpretation to a characteristic function of the form
$$
\varphi_{?}(t)=\frac{1}{1-i\lambda t}\mathbf{+}e^{i\mu t – \frac{1}{2}t^2\sigma^2}
$$
Best Answer
The definition of a characteristic function $$\varphi_X(t) =\mathrm{E}\left(e^{\mathrm{i}tX}\right),$$ implies $$\varphi_X(0) =\mathrm{E}1=1,$$ so $\varphi_X(t)+\varphi_Y(t)$ can't be a characteristic function. Of course, we can normalize it: $$\varphi_Z(t)=\frac12(\varphi_X(t)+\varphi_Y(t))=\frac12\mathrm{E}\left(e^{\mathrm{i}tX}\right)+\frac12\mathrm{E}\left(e^{\mathrm{i}tY}\right).$$ The probabilistic interpretation is obvious: you toss a fair coin, and "head" means $Z=X$, "tail" is $Z=Y$.