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If T is a complete sufficient statistic and δ is unbiased for g(θ), then a function of T is the unique uniformly minimum-variance unbiased estimator (UMVUE) of g(θ).

A boundedly complete sufficient statistic is independent of every ancillary statistic under each measure in the family.

For i.i.d. samples and a compact parameter space, continuous dominated criterion functions satisfy uniform almost-sure convergence of sample averages to population means.

A triangular array of independent, centered row-variables whose normalized sums satisfy the Lindeberg condition converges in distribution to the standard normal.

For independent centered Gaussian vectors with diagonal variances bounded above and away from zero, a covariance gap Δ in (0, 1) controls the difference between their maximum distributions.

The product of a probabilistically bounded sequence and a sequence converging in probability to zero also converges to zero in probability.

Little-o probability rates scale linearly with deterministic sequences, enabling rate transfer across normalizations.

Convergence in probability plus uniform integrability implies L¹ convergence.

For uniformly integrable sequences converging in probability, expectations converge.

A union-bound finite-class uniform deviation theorem for bounded measurable functions.

A finite-sample Glivenko-Cantelli theorem giving an explicit tail bound for the empirical distribution function.

A chaining-style entropy integral control for finite sub-Gaussian processes.

An L-Lipschitz function of a vector with independent standard Gaussian coordinates has a dimension-free two-sided Gaussian tail around its mean.

The square of a sub-Gaussian random variable is sub-exponential, with an explicit moment generating function bound.

The Gaussian Mehler semigroup is formalized as the analytic backbone of the functional-inequality and concentration chain.

For independent zero-mean sub-exponential variables, the tail interpolates between Gaussian and exponential decay depending on the deviation scale.

A function of independent variables that changes by at most c_i when one coordinate changes concentrates sharply around its mean.

A proved score-coverage event transfers its marginal coverage bound to any prediction event containing it.

A score miscoverage bound transfers to any smaller bad-prediction event.

The operator-convexity step that now replaces the former matrix Lieb trace axiom path.

The positive-definite simple-function Jensen step feeding the matrix Bernstein Laplace-transform argument.

For Ahat = A + E with a separated top-r singular spectrum and small operator-norm perturbation, the left and right singular subspace projectors move by at most a Wedin sin-theta bound.

Under rank-4r restricted isometry with δ below √2−1, a feasible nuclear-norm minimizer has Frobenius error bounded by the rank-r approximation tail and observation noise. The positive constants depend on δ.

For a positive-singular-value compact factorization M = U diag(σ) Vᵀ with orthonormal factor columns, characterizes the nuclear-norm subdifferential and supplies tangent-space certificate inequalities.

The matrix Bernstein bound extended to sums of independent centered rectangular p×q random matrices via Hermitian dilation, with variance the max of the two one-sided second-moment norms.

The isotropic specialization of Hanson-Wright, where the centering term is exactly the trace of A.

For i.i.d. centered sub-Gaussian vectors there is a universal constant so the sample second-moment matrix concentrates around the true covariance in operator norm with exponential tails.

An isotropic sub-Gaussian vector has squared norm tightly concentrated about its dimension n, with Bernstein-type sub-exponential tails.

A heterogeneous-covariance RIP concentration inequality: sub-Gaussian rows concentrate around their covariance quadratic form under a covariance lower bound κ.

An m-by-n matrix with i.i.d. isotropic sub-Gaussian rows satisfies the (s,delta) restricted isometry property with high probability once m exceeds order s·log(en/s)/delta^2.

A fixed-design LASSO oracle inequality stated through an L1 restricted-strong-eigenvalue condition.

An iid-score debiased LASSO theorem: row-approximation error, L1 consistency, studentization, and Gaussian critical-value calibration imply asymptotic standard Wald confidence-interval coverage.

Under the Restricted Eigenvalue condition RE(s,3,κ) and a noise control on λ, the in-sample prediction error of the LASSO is bounded by a sparsity-scaled multiple of λ².

Each sorted eigenvalue of a Hermitian matrix moves by at most the operator norm of the perturbation.

For a simple eigenvalue with eigengap Δ and a small symmetric perturbation, the perturbed unit eigenvector stays within 4‖E‖/Δ of the original.

An explicit fixed-width construction is converted into the architecture-scale ReLU approximation rate (L W^2)^(-2(r+beta)/d) for high-dimensional Holder-smooth functions.

A positive-degree tensor-product B-spline system on the high-dimensional unit cube achieves the uniform squared-error sieve rate m^(-2(r+beta)/d) over the trace Holder-smooth ball.

A measurable m-cell selector cover gives a piecewise-constant selector-indicator sieve with integrated squared-error rate m^{-2 alpha / d} over a Holder ball.