Stam inequality

Wiki card — concept node for exp-families-stability.

(i) Formal statement

Let \(X, Y\) be independent \(\mathbb{R}^d\)-valued random variables with smooth densities and finite Fisher information \(J(X), J(Y)\). Then the Stam (or Blachman–Stam) inequality asserts

\[ \boxed{\;\frac{1}{J(X + Y)} \;\ge\; \frac{1}{J(X)} \;+\; \frac{1}{J(Y)}\;} \tag{Stam} \]

with equality iff \(X, Y\) are jointly Gaussian of proportional covariance. A dimensional generalisation (for \(X \in \mathbb{R}^d\), isotropic Fisher info defined via \(\nabla\log p\)) holds in the same form.

Companion: the entropy-power inequality (EPI, Shannon 1948, rigorous proof Blachman 1965): \[ N(X + Y) \;\ge\; N(X) + N(Y), \qquad N(X) := \tfrac{1}{2\pi e}\exp\!\bigl(\tfrac{2}{d} h(X)\bigr). \]

Stam ⇒ EPI via integration of de Bruijn along the AWGN channel \(X_t = X + \sqrt{t}\, Z\); the route Stam → EPI is canonical.

(ii) Role in Q1 / Q2 / Q3

• Q1 (Shannon-obstruction chain). The pair (de Bruijn, Stam) is the information-theoretic footprint of the heat flow. Stability of \(\mathcal{E}_\varphi\) under \(*\gamma_\sigma\) requires Fisher information and entropy along the flow to be affine in moments of \(\varphi\); Stam is the inequality that saturates exactly when \(X, Y\) are Gaussian, the Q1-obstruction to non-Gaussian stability.
• Q2 (quadratic). Gaussian \(\mathcal{N}(\mu, \Sigma)\) saturates Stam, hence the Q2 reparameterisation corresponds to the equality case of the Blachman–Stam inequality, a structural coincidence.
• Q3 (HJ integrated). Stam is the convex-analytic dual of the Hamilton–Jacobi equation: both express that "along the heat flow, Fisher info behaves as \(t \mapsto 1/(c + t)\)". See the OU worked miniature below.

(iii) References

• Stam (1959) — Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon. doi:10.1016/S0019-9958(59)90348-1.
• Blachman (1965) — The Convolution Inequality for Entropy Powers. doi:10.1109/TIT.1965.1053768.
• Dembo, Cover, Thomas (1991) — Information-Theoretic Inequalities. doi:10.1109/18.104312.
• Shannon (1948) — A Mathematical Theory of Communication (original EPI). doi:10.1002/j.1538-7305.1948.tb01338.x.

(iv) Worked miniature — Gaussian saturation

Take \(d = 1\), \(X \sim \mathcal{N}(0, a)\), \(Y \sim \mathcal{N}(0, b)\) independent. Then \(X + Y \sim \mathcal{N}(0, a + b)\).

• \(J(X) = 1/a\), \(J(Y) = 1/b\), \(J(X+Y) = 1/(a+b)\).
• RHS of Stam: \(\tfrac{1}{J(X)} + \tfrac{1}{J(Y)} = a + b\).
• LHS of Stam: \(\tfrac{1}{J(X+Y)} = a + b\).

Equality. ✓ This shows the Gaussian family is exactly the saturation locus of Stam, and correspondingly the Q2 stable class.

Dual proof via de Bruijn (Blachman 1965 sketch). For \(X, Y\) with Fisher infos \(J(X), J(Y)\), take \(Y = \sqrt{\lambda}\, Z\) with \(Z \sim \mathcal{N}(0, I)\) and let \(\lambda\) run. Use de Bruijn to compute \(\tfrac{d}{d\lambda}\) of both sides of an inequality \(\lambda + 1/J(X) \ge 1/J(X + \sqrt\lambda Z)\) and show the RHS derivative is \(\le 1\) (Cauchy–Schwarz on \(\nabla\log p_{X + \sqrt\lambda Z}\)). Integration over \(\lambda \in [0, b]\) gives Stam in the form stated.

Implication for stability. Along the heat flow \(X_t = X + \sqrt{t}\,Z\), \(J(X_t)\) is bounded above by \(J(X + \sqrt{t}\,Z) \le 1/t\) via Stam with \(Y = \sqrt{t}\,Z\) (Fisher info \(= 1/t\)). The upper bound is attained iff \(X\) itself is Gaussian — again the Q2 saturation.