Showing a Random Variable is Discrete from the Characteristic Function

Question

Question:

The characteristic func. of a r.v. X is $$\dfrac{e^{it}(1-e^{nit})}{n(i-e^{it})}$$ Show that X is a discrete r.v with $p(x)=\dfrac1n$ for $x=1,2,\cdots n.$

Can one please help me to show the if X random variable is discrete using the characteristic function? I couldn't solve and actually I didn't understand how to solve.

Answer 1

In this case, note $\frac{1-z^n}{1-z}=\sum_{k=0}^{n-1}z^k$, so the characteristic function is $\frac1n\sum_{j=1}^ne^{jit}$.

Answer 2

The general question, of determining if a random variable $X$ is discrete by examining its characteristic function $\varphi_X$, is hard. Lukacs's Characteristic Functions discusses this on pp. 18-20. If there exists a $t\ne0$ such that $|\varphi_X(t)|=1$, then $X$ is a lattice random variable and hence discrete. Define $L$ by $$L=\limsup_{|t|\to\infty}|\varphi_X(t)|.$$ If $X$ is discrete, then $L=1$, so if $L<1$ then $X$ is not discrete. But if $L=1$ the most one can conclude in general is that the Lebesgue decomposition of the distribution of $X$ into absolutely continuous, singular continuous, and discrete components contains no absolutely continuous component. But the distribution of $X$ might be discrete or might be singular, or might be a mixture of the two.