[Paper Review] Convergence of the Poincare Constant

Oliver Johnson|Jun 21, 2002

Mathematical Dynamics and Fractals9 references3 citations

TL;DR

This paper establishes the best possible rate of convergence for the Poincaré constant toward 1 in the Central Limit Theorem, proving that $ R_{U_n} - 1 \leq C/n $ for normalized sums $ U_n $, with $ C $ depending only on the initial Fisher information and Poincaré constant. It further shows that discrete random variables perturbed by small normals have finite Poincaré constants, enabling explicit convergence rates in relative entropy for the CLT.

ABSTRACT

The Poincare constant R(Y) of a random variable Y relates the L2 norm of a function g and its derivative g'. Since R(Y) - Var(Y) is positive, with equality if and only if Y is normal, it can be seen as a distance from the normal distribution. In this paper we establish a best possible rate of convergence of this distance in the Central Limit Theorem. Furthermore, we show that R(Y) is finite for discrete mixtures of normals, allowing us to add rates to the proof of the Central Limit Theorem in the sense of relative entropy.

Motivation & Objective

• To determine the best possible rate of convergence of the Poincaré constant to 1 in the classical Central Limit Theorem.
• To show that discrete random variables perturbed by small independent normals have finite Poincaré constants, enabling quantitative convergence results.
• To provide explicit rates of convergence in relative entropy for the CLT using the Poincaré constant as a tool.
• To establish a sharp bound on the decay of $ R_{U_n} - 1 $, showing it is $ O(1/n) $, and that this rate is optimal up to the constant.

Proposed method

• Uses the Poincaré constant $ R_Y $ as a spectral gap measure, defined as the supremum of $ \mathrm{Var}(g(Y)) / \mathbb{E}[g'(Y)^2] $ over suitable functions $ g $.
• Applies subadditivity of the Poincaré constant: $ R_{X+Y} \leq R_X + R_Y $ for independent $ X,Y $, and uses this to control the decay along the $ 2^k $-subsequence.
• Employs a variational characterization of the Poincaré constant via eigenfunctions of the score-driven Laplacian operator $ D_Y g = \rho_Y g' + g'' $, linking it to Fisher information.
• Uses a perturbation argument: for any discrete $ X $, $ Y_\tau = X + Z_\tau $ with $ Z_\tau \sim N(0,\tau) $ has a finite Poincaré constant bounded by $ \tau \left(1 + \left(\frac{\sigma^2}{\tau \min_s p_s}\right) \exp\left(\frac{\sigma^2}{\tau \min_s p_s}\right)\right) $.
• Applies Lemma 3.2 to bound the deviation $ R_{U_n} - 1 $ in terms of the variance of the derivative of a test function, using a local maximization argument.
• Uses recursive inequalities on the $ 2^k $-subsequence to derive $ u_{2^r} \leq 4 / 2^r $, then extends to full sequence via subadditivity to show $ R_{U_n} - 1 = O(1/n) $.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence of the Poincaré constant $ R_{U_n} $ to 1 as $ n \to \infty $ in the classical Central Limit Theorem?
• RQ2 Can the Poincaré constant be finite for discrete random variables, and if so, what is a sharp bound for its value after convolution with small Gaussian noise?
• RQ3 Does the finiteness of the Poincaré constant imply explicit rates of convergence in relative entropy for the CLT?
• RQ4 Is the $ O(1/n) $ rate of decay of $ R_{U_n} - 1 $ sharp, and can it be proven via subadditivity and recursive inequalities?

Key findings

• The Poincaré constant $ R_{U_n} $ of the normalized sum $ U_n = (X_1 + \cdots + X_n)/\sqrt{n\sigma^2} $ satisfies $ R_{U_n} - 1 \leq C/n $, where $ C $ depends only on the initial Fisher information $ I $ and Poincaré constant $ R $, proving the best possible rate up to the constant.
• For any discrete random variable $ X $ with finite support and variance $ \sigma^2 $, the perturbed variable $ Y_\tau = X + Z_\tau $ with $ Z_\tau \sim N(0,\tau) $ has a finite Poincaré constant bounded by $ \tau \left(1 + \left(\frac{\sigma^2}{\tau \min_s p_s}\right) \exp\left(\frac{\sigma^2}{\tau \min_s p_s}\right)\right) $.
• The convergence of $ R_{U_n} $ to 1 is sharp in the sense that $ R_{U_n} - 1 $ cannot decay faster than $ O(1/n) $, as shown by a counterexample using $ g(x) = x^2 - 1 $ and the fourth moment.
• The finiteness of $ R_Y $ for $ Y = X + Z_\tau $ implies that the relative entropy distance to the normal distribution decays at a controlled rate, enabling strong convergence results in $ L^1 $ and for expectations of sub-exponential functions.
• The proof uses a recursive inequality on the $ 2^k $-subsequence: $ (R_{S_{k+1}} - 1)^2 \leq C(R_{S_{k+1}} - R_{S_k}) $, which leads to $ R_{U_n} - 1 = O(1/n) $ after filling in the gaps via subadditivity.
• The result implies that for any continuous function $ w $ with $ |w(t)| \leq \exp(c|t|) $ for $ c < 1/(12\sqrt{R}) $, $ \mathbb{E}[w(U_n)] \to \mathbb{E}[w(Z)] $, showing convergence in distribution with exponential moment control.