characteristic function upper bound and uniformly continuous.

Question

Let $X$ be a random variable and let $\phi$ be its characteristic function. Let $A$ be a nonnegative constant and consider the following inequality $$ |\phi(t)-\phi(s)| \leq \sqrt{A|1-\phi(t-s)|}. $$ Show that if the inequality holds for all random variables then $A\geq 2$. Specifically, show that for any $A< 2$ there is an $X$ for which the inequality fails.

I am not quite sure about how to use the uniform continuity to prove this conclusion.

Answer 1

I am not quite sure about how to use the uniform continuity to prove this conclusion.

Then don't.

Maybe we should apply some techniques in Fourier analysis.

Hmmm... And maybe we should look for explicit simple counterexamples?

Try $X$ such that $P(X=1)=1$, and $s=0$, $t\to0$.