AI.doll

このブログは僕のためのメモです。

$$ \newcommand{\inner}[2]{\langle #1, #2 \rangle} \newcommand{\matr}[1]{\boldsymbol{#1}} \newcommand{\pdif}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pdifn}[3]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\dif}[2]{\frac{d #1}{d #2}} \newcommand{\bracket}[1]{\left\langle #1 \right\rangle} $$

Lyapunov inequality

確率不等式

Lyapunov不等式

s>r>1のとき, \left\langle|X|^r\right\rangle^{1/r}\leq\left\langle|X|^s\right\rangle^{1/s}.

証明

Hölderの不等式で Y=1とすると,

\begin{aligned} \left\langle|X|^r|1|\right\rangle\leq\left\langle|X|^{rp}\right\rangle^{1/p}\left\langle|1|^q\right\rangle^{1/q}, \frac{1}{p}+\frac{1}{q} = 1. \end{aligned}

p=\frac{s}{r}とできるので,

\begin{aligned} \left\langle|X|^r\right\rangle \leq \left\langle|X|^s\right\rangle^{r/s}. \end{aligned}

両辺を 1/r乗すると,

\begin{aligned} \left\langle|X|^r\right\rangle^{1/r} \leq \left\langle|X|^s\right\rangle^{1/s}. \end{aligned}