Limitation on the difference of characteristic functions

Question

Let $ X, Y $ be independent with characteristic functions $\varphi_X(t) , \varphi_Y(t) $.

Show that: $$ \sup_{t \in \mathbb{R}}\mid \varphi_x(t) - \varphi_Y(t) \mid \le 2P(X \neq Y) $$

I would appreciate any tips or hints.

Answer 1

Note that for all $t,x,y\in \mathbb{R}$, it holds $$ |e^{itx}-e^{ity}|\leq 2 \cdot 1_{\{x\neq y\}}. $$ Thus, for any $t\in \mathbb{R}$, we have $$ |\varphi_X(t)-\varphi_Y(t)|\leq E[|e^{itX}-e^{itY}|]\leq 2E[1_{\{X\neq Y\}}]=2P(X\neq Y). $$ Now, take supremum over $t\in \mathbb{R}$ to get $$ \sup_{t\in\mathbb{R}}|\varphi_X(t)-\varphi_Y(t)|\leq 2P(X\neq Y), $$as desired.

Answer 2

What sort of distributions do $X$ and $Y$ have? If the distributions have no discrete terms, $P(X\ne Y)=1$ and the inequality always hold trivially, since all $|\phi(t)|\le 1.$