Let $U_1, \ldots, U_n$ be uncorrelated random variables with zero mean and finite variance. Define $X = \sum_{i=1}^n U_i$ and $X_m = \sum_{i=1}^m U_i$ for $m \le n$. Let $\varphi_X(t)$ and $\varphi_m(t)$ denote the characteristic functions of $X$ and $X_m$, respectively. Then, is it true that $| \varphi_X(t) - \varphi_m(t) | \le |t| \{ E[ (X - X_m)^2 ] \}^{1/2}$?
I don't get how this follows, or how to prove it. This is used in the proofs in a statistics paper without reference. Will this statement be true for any general random variables $X$ and $Y$ with finite second moments?
We have: $$|\varphi_X(t) - \varphi_{X_m}(t)| \leq \mathbb{E}\left( \left|e^{itX} - e^{itX_m}\right| \right)$$ Now, as the derivative of $f_t: x \mapsto e^{itx}$ is $x \mapsto ite^{itx}$ bounded by $|t|$, $f_t$ itself is $|t|$-Lipschitz (why?), hence: $$\left|e^{itx} - e^{itx'}\right| \leq |t| |x - x'|$$ which then provides: $$\mathbb{E}\left( \left|e^{itX} - e^{itX_m}\right| \right) \leq |t| \mathbb{E}\left( \left|X - X_m\right| \right) \leq |t|\mathbb{E}\left( \left(X - X_m\right)^2 \right)^{1/2}$$ where the last inequality is just Cauchy-Schwarz.