A Dimension-Free Central Limit Theorem for Log-Concave Inner Products via Stein Kernels

Source abstract

Let $X,Y\in\mathbb R^n$ be independent isotropic log-concave random vectors. Using the quadratic-form and moment-map matrix-energy estimates proved in Theorems 1.2 and 2.5 of [Let26], we derive a covariance bound for trace observables of the moment-map Stein kernel: $$\operatorname{Var}\!\left(\operatorname{Tr}(Aτ(Y))\right) \leq 4\operatorname{Tr}(A^2),\qquad A=A^{\mathsf T}.$$ The main ingredient, in the regular moment-map model, is the exact identity $$\operatorname{Var}\!\left(\operatorname{Tr}(Aτ(Y))\right) =\operatorname{Var}(Y^{\mathsf T}AY) -3\mathbb{E}(AY)^{\mathsf T}τ(Y)(AY) +\mathbb{E}\operatorname{Tr}(Aτ(Y)Aτ(Y)).$$ Combining this trace bound with conditional quadratic-form control produces a scalar Stein kernel for $\langle X,Y\rangle$ whose normalized squared $L^2$ discrepancy is at most $20/n$. Consequently, $$W_2^2 \left( L\!\bigl(\frac{\langle X,Y\rangle}{\sqrt n}\bigr),N(0,1) \right)\leq\frac{20}{n}, \qquad W_2^2\!\left(L(\langle X,Y\rangle),N(0,n)\right)\leq20.$$ This proves the dimension-free central limit theorem proposed by [JLV20], subject to the cited preprint results. We include cutoff and approximation details, exact dependency accounting, and Gaussian and product-exponential checks.