Welcome to my blog!

This is the first post of this blog as well as a testing post.

MathJax testing

Karamata's proof of Hardy-Littlewood Tauberian theorem can be easily adapted to yield a proof the following proposition:

Proposition. Let $X$ be a non-negative random variable and $\mathcal{L}_X (s) = \mathbb{E}[e^{-sX}]$ be its Laplace transform. If $A > 0$ and $b \geq 0$, then
$$ \mathcal{L}_X(s) \sim \frac{A}{s^b} \text{ as } s \to \infty \quad \Longleftrightarrow \quad F_X(\epsilon) \sim \frac{A\epsilon^{b}}{\Gamma(b+1)} \text{ as } \epsilon \to 0^+. \tag{*}$$

Language testing

• 한글은 잘 보입니까?
• ひらがな、カタカナ、漢字、皆よく見えますか？