Lyapunov Inequality
1 Statement
Lyapunov’s inequality says that the \(L^r\) norms of a random variable are increasing in \(r\). It is a special case of the Hölder Inequality, and explains why higher-order [[convergence-in-lp-norm|\(L^s\) convergence]] implies lower-order \(L^r\) convergence.
For a random variable \(X\) and \(0 < r < s < \infty\), \[ (\mathbb{E}[\lvert X \rvert^r] )^{1/r}\leq (\mathbb{E}[\lvert X \rvert ^s])^{1/s}. \]
Proof. Hölder’s inequality states that for \(p,q>1\) with \(1/p+1/q=1\) \[ |\mathbb{E}[UV]|\leq \mathbb{E}|UV|\leq (\mathbb{E}|U|^p)^{1/p}\cdot (\mathbb{E}|V|^q)^{1/q}. \] Take \(U=\lvert X \rvert^r\), \(V=1\) and \(p=s/r>1\). Then \[ \mathbb{E}\lvert X \rvert ^r\leq (\mathbb{E}\lvert X \rvert ^{rp})^{1/p} = (\mathbb{E}\lvert X \rvert^s )^{r/s}, \] and raising both sides to the power \(1/r\) gives the result.