Proof of Lyapunov's Inequality

About Posts which Tagged by 'Probability'

Lyapunov's inequality.  Let $X$ be a random variable.  For $0<s<t$, $$\left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t}.$$
See List of Inequalities.

$\bullet$ Proof.

(1) For $1\leq s<t$, set $Y=1$ in H$\ddot{o}$lder's inequality, we have $$\mathscr{E}|X|\leq\left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}.$$Let $t=sp$ and replace $|X|$ by $|X|^s$, we, have $$\begin{array}{rl} & \mathscr{E}|X|^s\leq\left(\mathscr{E}|X|^{sp}\right)^{\frac{1}{p}}\\
\implies & \mathscr{E}|X|^s\leq\left(\mathscr{E}|X|^{t}\right)^{\frac{s}{t}}\\
\implies & \left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t} \end{array}$$
(2) For $0<s<t<1$, define $r=\frac{t}{s}>1$ and $Y=|X|^s$.  Let $\phi(y)=y^r$, thus $\phi$ is a convex function.  Then by Jensen's inequality, we have $$\begin{array}{rl} & \phi(\mathscr{E}Y)\leq\mathscr{E}[\phi(Y)] \\ \implies
& \left(\mathscr{E}|X|^s\right)^r\leq\mathscr{E}|X|^{sr}\\ \implies
& \left(\mathscr{E}|X|^s\right)^{\frac{t}{s}}\leq\mathscr{E}|X|^{t}\\ \implies
& \left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t}. \end{array}$$

$\Box$