Lyapunov moment inequality
= Lyapunov moment inequality
{c}
{title2=$\|Z\|_r\leq\|Z\|_s\quad(0<r<s)$}
For a <random variable> $Z$ and exponents $0<r<s$, the absolute <moment> norms satisfy $(\mathbb E|Z|^r)^{1/r}\leq(\mathbb E|Z|^s)^{1/s}$, allowing infinity. For a finite $s$th <moment>, apply the <Jensen inequality> to the <concave function> $t^{r/s}$ and $|Z|^s$. The formula is for positive exponents; $r=0$ requires a separately defined limiting quantity.