# Characteristic function of bounded random variables

Does characteristic function of bounded random variables have any additional properties?

More specifically, let $X$ be a symmetric random variable such that $P[|X|\le a]=1$.

What more can we say about $\phi_X(t)$ ?

• When you say "additional properties", in addition to what do you have in mind? – grndl Aug 24 '16 at 15:56
• In addition to what we already to know about characteristic function: continuity, $\phi(0)=1$. For example, can we say something about the tails of $\phi(t)$. – Boby Aug 24 '16 at 16:00
• – Carlo Beenakker May 28 at 20:43

The boundedness of $X$ implies that moment generating function $\psi(z):=\Bbb E[e^{zX}]$ is an entire function $\psi:\Bbb C\to\Bbb C$. Of course $\phi_X(t) = \psi(it)$ for all real $t$. In particular, $\phi_X$ is a smooth function. The symmetry of $X$ implies (and is indeed equivalent to the fact) that $\phi_X$ is real valued.
• Would characteristic function have zeros if $X$ is continuous? What if it's discrete? – Boby Aug 24 '16 at 23:29
• Examples: $X$ uniformly distributed on $(-1,1)$, hence bounded and symmetric. Then $\phi_X(t) = {\sin t\over t}$ for $t\not=0$ and $=1$ for $t=0$. This characteristic function has many zeros. Similarly, if $X$ takes the two values $\pm1$ with probability $1/2$ each, then $\phi_X(t) =\cos t$. – John Dawkins Aug 25 '16 at 16:37