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, not a miracle.
• 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.