Section 14.3

Bounds on Moments and Correlations

Positive-semidefinite constraints on correlations, the largest variance on an interval, Frechet bounds, inequalities for moment generating functions.

Toolkit

These problems exploit three structural constraints: boundedness, positive-semidefiniteness, and convexity.

Variance on an interval. If $X\in[a,b]$ , then $\operatorname{Var}(X)\le \frac{(b-a)^2}{4}$ , maximised by a two-point distribution at the endpoints.

Fréchet bounds. For events or random variables,

\max(\mathbb P(A)+\mathbb P(B)-1,0)\le \mathbb P(A\cap B)\le \min(\mathbb P(A),\mathbb P(B)),

with equality cases given by nested or complementary configurations.

Correlation matrices are PSD. Any valid correlation matrix $R$ satisfies $R\succeq 0$ ; for $n$ equally correlated variables this forces $\rho\ge -1/(n-1)$ (from $\operatorname{Var}(\sum X_i)\ge0$ ), and for three variables,

\rho_{13}\in\rho_{12}\rho_{23}\pm\sqrt{(1-\rho_{12}^2)(1-\rho_{23}^2)}.

Jensen’s inequality. For convex $\varphi$ , $\mathbb E[\varphi(X)]\ge \varphi(\mathbb E[X])$ ; applied to $\varphi(x)=e^{tx}$ gives lower bounds on moment generating functions, with equality iff $X$ is constant.

Problems