= Lyapunov moment inequality
{c}
{title2=$\|Z\|_r\leq\|Z\|_s\quad(0<r<s)$}
The Lyapunov moment inequality says that higher positive <absolute moments> give no smaller <absolute moment> sizes on a <probability space>. For a real <random variable> $Z$ and $0<r<s$,
$$
\boxed{\left(\mathbb E|Z|^r\right)^{1/r}\leq\left(\mathbb E|Z|^s\right)^{1/s},}
$$
allowing an infinite value. For $r\geq1$ the left expression is an <Lp norm>; below one it is a <quasi-norm>, so the usual <triangle inequality> terminology needs that distinction. The comparison uses a <probability measure> of total mass one; arbitrary unnormalized <measure spaces> need a factor depending on their total mass.
For a nonnegative count, the first moment size is its <expected value>, while the second is the <square root> of its <second moment>. Both ten-trial counts in <second moment method> have first moment size one. The <independent> count has second moment size $\sqrt{1.9}\approx1.37840$, whereas the shared-outcome count $N=10B$ has second moment size $\sqrt{10}\approx3.16228$. The second moment distinguishes models with the same average because its square increases the influence of rare large counts.
More generally, for $N=10B$ with a <Bernoulli random variable> $B$ of parameter $1/10$,
$$
\left(\mathbb E N^r\right)^{1/r}=10(1/10)^{1/r}.
$$
It rises from one at $r=1$ toward ten as $r\to\infty$. Replacing $1/10$ by a parameter $0<q\leq1$ gives $10q^{1/r}$, which decreases toward zero for a fixed positive <exponent> when $q\downarrow0$, making the rare-event and large-exponent limits different questions. For a scaled <Rademacher random variable> taking values $-2$ and two with equal <probabilities>, every absolute moment size is exactly two: increasing the <exponent> does not distinguish signs when the <absolute value> is constant.
\Image[/lyapunov-moment-sizes-for-rare-counts.png]
{title=<Lyapunov moment inequality> reveals how higher <moments> emphasize large outcomes and when equality holds}
{description=Left: the two original ten-trial models from <second moment method> have the same <mean> one, but different <absolute moment> sizes at <exponents> above one. The shared model has $N=10B$ with $p=1/10$. Right: equal masses at $-2$ and two keep $|Z|=2$ <almost surely>, giving equality for every pair of positive <exponents>. These are fully specified mathematical probability models.}
For finite $\mathbb E|Z|^s$, apply the reversed <Jensen's inequality> to the <strictly concave function> $t^{r/s}$ and the nonnegative <random variable> $|Z|^s$:
$$
\mathbb E|Z|^r\leq\left(\mathbb E|Z|^s\right)^{r/s}.
$$
Taking the positive $r$th root proves the claim. Equality in this finite case holds exactly when $|Z|$ is constant <almost surely>. If the higher <moment> is infinite, the upper bound is infinite and remains valid. Exponent zero is excluded from the statement; logarithmic or geometric limiting summaries require separately stated conventions, especially when $Z$ can vanish.
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