Infinite divisibility of a binomial random variable

Question

I am trying to figure whether or not the binomial distribution is infinitely divisible. As binomial random variables are bounded I believe that they are not infinitely divisible, but I am unsure how to prove this properly.

Answer 1

The only infinitely divisible distributions with compact support are the degenerate ones. In particular if $X$ takes only finite number of values and if $X$ is not a constant then it cannot be infinitely divisible.

No bounded random variable is infinitely divisible. Proof: Suppose $|X| \leq M$ and $X$ is infinitely divisible. For any $n$ we can find i.i.d. random variables $X_1,X_2,...,X_n$ such that $X_1+X_2+...+X_n$ has the same distribution as $X$. We have $0=P(X>M)\geq P(X_1 >\frac M n,X_2 >\frac M n,...X_n >\frac M n) =(P(X_1 >\frac M n))^{n}$ so $X_1 \leq \frac M n$ almost surely. Similarly $X_1 \geq -\frac M n$ almost surely. So $|X_1| \leq \frac M n$ almost surely and $var (X_1) \leq \frac {M^{2}} {n^{2}}$. Since $var (X)=nvar(X_1)$ we see that $var X \leq \frac {M^{2}} n$. This is true for any $n$ so $var (X)=0$. This implies that $X$ is a constant.