derivative of characteristic function I came across an interesting problem but unable to see how to approach it. How do I use the dominated convergence theorem (LDCT), to show that first derivative of the characteristic function of the probability distribution at $t = 0$, $\phi^′(0)=iE[X]$? Any ideas?
References:
http://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
http://en.wikipedia.org/wiki/Dominated_convergence_theorem

You need to show that if $(a_n)$ is a sequence of nonzero numbers
which tends to zero, that $(\phi(a_n)-\phi(0))/a_n\to iE[X]$. Now
$$\frac{\phi(a)-\phi(0)}{a}=E\left(\frac{e^{iaX}-1}{a}\right).$$
You need some hypothesis on $X$ for the result to work, for
instance that $|X|$ has finite expectation. Certainly $(e^{iaX}-1)/a\to iX$
so to apply dominated convergence we need a function $f(X)$ with finite
expectation and with $|(e^{iaX}-1)/a|\le f(X)$ at least for $a$ in a
deleted neighbourhood of $0$. Does $f(X)=|X|$ work?
