Mathematics Asked by Filippo Giovagnini on December 12, 2020

**Background:**

I am trying to understand why, if $X$ is a random variable with $mathbb{E}big[|X|big]< +infty$ and $mu=mathbb{P}^{X}$, then $$frac{partial}{partial x_j}overline{mu}(u)=iint e^{i langle u,x rangle} x_j mu(dx).$$

**My problem:**

Suppose $t_p to 0$ and $x in mathbb{R}^n$. How can I prove that $$left|frac{e^{it_p x_j}-1}{t_p}right| leq 2|x|,$$ where $x_j$ is the $j$-th coordinate of $x$?

In fact you can get $|x|$ as a bound. Use the fact that $|e^{itheta} -1| leq |theta|$ for all real numbers $theta$.

[$int_0^{1} e^{ittheta} dt=frac 1 {i theta} (e^{itheta}-1)$ and LHS is bounded in modulus by $int_0^{1}dt=1$].

Correct answer by Kavi Rama Murthy on December 12, 2020

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