Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Weighted Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space
definitiondef:weighted-fisher-information-euclidean-2026aProbabilityFor a measure on Euclidean space with finite Fisher information relative to a diagonal Gaussian and positive weights, defines the weighted Fisher information as the integral of the weighted sum of squares of the components of the relative score.Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space
definitiondef:weighted-fisher-information-hilbert-2026aProbabilityDefines weight sequences (bounded sequences of positive numbers) and the Fisher information relative to a diagonal Gaussian with given weights: the weighted sum of the squared L^2 norms of the score components, when it converges.Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs
lemmalem:relative-score-projections-hilbert-2026aProbabilityFor a measure on a Hilbert space with a relative score, the finite-dimensional projections have finite Fisher information and their scores are the orthogonal projections of the score components onto functions of the first n coordinates; the weighted cutoff Fisher informations inc…Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded Functions
lemmalem:relative-score-componentwise-euclidean-2026aProbabilityOn Euclidean space, finite Fisher information relative to a diagonal Gaussian, defined through the tangent space, is equivalent to componentwise Gaussian integration by parts against bounded C^1 functions; the components are then unique and equal to those of the relative score, a…The Relative Score on a Hilbert Space: the Gaussian Measure Has Score Zero, and the Score as a Square-Integrable Field in the Weighted Sequence Space
lemmalem:relative-score-basic-hilbert-2026aProbabilityThe diagonal Gaussian measure has relative score zero and zero weighted Fisher information; and a measure with finite weighted Fisher information has a unique score field in the square-integrable maps into the weighted sequence space, whose squared norm is the Fisher information.The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space
definitiondef:relative-score-diagonal-gaussian-hilbert-2026aProbabilityA probability measure with finite second moment on a Hilbert space has a relative score with respect to a diagonal Gaussian when, for each coordinate, Gaussian integration by parts against bounded C^1 cylindrical functions is represented by a square-integrable function; the score…The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis
lemmalem:square-integrable-maps-hilbert-valued-2026aProbabilityThe space of square-integrable Hilbert-valued maps is a real Hilbert space; its inner product is the sum of the scalar L^2 inner products of the coordinates, and every square-summable sequence of scalar classes is the coordinate sequence of exactly one element.The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis
definitiondef:square-integrable-maps-hilbert-valued-2026aProbabilityDefines the space of almost-everywhere classes of square-integrable maps from a measure space into a Hilbert space with an orthonormal basis, with pointwise operations and the integrated inner product.Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality
lemmalem:measurable-maps-hilbert-valued-2026aProbabilityMeasurable maps into a Hilbert space with an orthonormal basis are closed under linear combinations and have measurable norms, inner products and coordinates; square-summable measurable coordinates synthesise a measurable map; square-integrable maps have integrable inner products…Relative Entropy with Respect to a Diagonal Gaussian Measure on a Hilbert Space: the Moment Bound, the Cutoff Projections, and Bounded, Tight, Weakly Closed, Wasserstein-Closed and Weakly Sequentially Compact Sublevel Sets
propositionprop:relative-entropy-diagonal-gaussian-hilbert-2026aProbabilityRelative to a diagonal Gaussian measure on a Hilbert space, finite relative entropy implies finite second moment with a linear bound, the entropy is the limit of the entropies of the finite-dimensional cutoffs, and entropy sublevel sets are bounded, tight, weakly closed, Wasserst…Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation
lemmalem:relative-entropy-projections-hilbert-2026aProbabilityFor Borel probability measures on a Hilbert space with an orthonormal basis, the relative entropies of the projections to the first n coordinates are nondecreasing in n, and the relative entropy is finite exactly when they are bounded, in which case it is their limit.Relative Entropy on a Metric Space: the Variational Criterion over Bounded Lipschitz Functions and Sequentially Closed Sublevel Sets under Weak Convergence
lemmalem:relative-entropy-weak-closed-metric-2026aProbabilityOn a metric space it suffices to test the variational criterion for relative entropy against bounded Lipschitz functions; consequently sublevel sets of the relative entropy are sequentially closed under weak convergence.Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing
lemmalem:relative-entropy-variational-measurable-2026aProbabilityOn any measurable space, relative entropy dominates the Donsker-Varadhan functional of every bounded measurable test function and equals its supremum; a uniform bound on that functional forces finite relative entropy. Consequences: the entropy inequality, a bound on the mass of s…- A variance sequence is a sequence of positive reals with convergent series; its truncation to the first n terms is a variance vector, with the associated diagonal Gaussian measure on n-dimensional Euclidean space.
Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space
theoremthm:gaussian-integration-by-parts-hilbert-2026aProbabilityFor a function of the first n coordinates with a gradient of at most linear growth, the integral of a coordinate derivative against a diagonal Gaussian measure equals the integral of the function times the coordinate divided by its variance, and the derivative along a Cameron-Mar…The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space
theoremthm:cameron-martin-translation-hilbert-2026aProbabilityTranslating a diagonal Gaussian measure by a Cameron-Martin vector h produces the measure with density exp of the Paley-Wiener functional of h minus half the Cameron-Martin square of h.The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square
lemmalem:paley-wiener-functional-hilbert-2026aProbabilityThe Paley-Wiener functional is Borel and equals the scaled coordinate for basis vectors; for a Cameron-Martin vector its partial sums converge almost everywhere and in mean square under the diagonal Gaussian measure, and its mean is zero and its second moment the Cameron-Martin s…The Paley-Wiener Functional of a Vector Relative to a Variance Sequence
definitiondef:paley-wiener-functional-hilbert-2026aProbabilityFor a variance sequence and a vector h, the Paley-Wiener partial sums are the finite sums of the coordinates of h times those of x divided by the variances, and the Paley-Wiener functional is their limit where it exists and zero elsewhere.The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space
definitiondef:cameron-martin-space-hilbert-2026aProbabilityThe Cameron-Martin space of a variance sequence consists of the vectors whose squared coordinates divided by the variances form a convergent series, and the Cameron-Martin square is the sum of that series.Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm
lemmalem:diagonal-gaussian-measure-moments-hilbert-2026aProbabilityUnder a diagonal Gaussian measure the coordinates are centred and uncorrelated with variances given by the variance sequence, the second moment is the sum of the variances, and the squared norm has exponential moments below the threshold set by the largest variance.