A mathematical preprint has identified a sharp dividing line between covariance-based formulas whose denominators can be calibrated without knowing the underlying covariance matrix and formulas that can become unstable. In its fixed-dimensional Gaussian setting, a nonconstant polynomial denominator is self-normalizing exactly when it has an exact relative pivot and a flag-power form, an algebraic pattern built from nested covariance blocks. In ordinary terms, that structure makes the sample-to-population ratio follow a finite-sample law that does not depend on the unknown covariance matrix. The exact law, however, is a Gaussian-reference result.
A rigid algebraic pattern
The main analytic setup used independent Gaussian covariance samples with an unknown positive-definite population covariance. Within that experiment, the paper studies polynomial denominators in covariance-based ratios and asks whether their first-order relative variance can be independent of the covariance. In other words, it tests whether denominator behavior is structural or tied to the particular covariance matrix.
The classification is exact. For a nonconstant covariance polynomial, self-normalization, a relative pivot and flag power are equivalent properties. The same result says that constant positive first-order relative variance forces a covariance-independent finite-sample law for the sample-to-population ratio. Exact calibration therefore follows from the denominator’s algebraic form, not from a particular covariance value.
A broader notion, relative stability, covers cases without that exact pivot. The paper identifies it with uniform parametric-rate control of relative denominator errors under Gaussian sampling. In plain language, the denominator can remain random, but its proportional error is controlled at the usual sample-size rate across the relevant covariance values. This is broader than exact covariance-free pivoting and is the condition linked to ratio limit theory.
When positivity is not enough
The boundary analysis becomes more specific for products of covariance minors, determinant factors built from covariance matrices. In that class, relative stability holds exactly when every factor is principal. A first-order weak-denominator sequence appears exactly when a nonprincipal factor occurs once. The result also says that a positive polynomial cannot produce a first-order weak sequence at a positive-semidefinite boundary zero: its first-order standard deviation is zero there.
The Napkin example shows why positivity alone is not enough. Its denominator stays positive everywhere but is still relatively unstable. The boundary behavior is non-Gaussian and is driven by the residual variance, which the authors classify as a higher-order form of weakness rather than first-order weakness.
Different formulas, different risks
Conditional instrumental-variable formulas and scalar linear proximal formulas have almost-principal, multiplicity-one denominators, the pattern the paper places in the first-order weak regime. Regression along one fixed coordinate order behaves differently: it does not produce a relatively unstable denominator, regardless of the polynomial target in the recursive regression coefficients.
The mediation and front-door product is classified as a self-normalizing denominator, with a self-normalization constant of 10. In a separate result on variance collapse, the Wald statistic using an empirical standard error remains asymptotically standard normal along every bounded positive sequence of mediator residual variances, provided the treatment to mediator coefficient is fixed and nonzero and the stated score-covariance condition holds. When that score covariance is the identity, the Gaussian-reference version also holds.
The finite-sample test
Numerical experiments tested these distinctions in finite samples. Each design cell used 50,000 independent samples, and intervals were set at nominal 95% coverage. The reported Monte Carlo standard error for coverage, a measure of simulation noise, was about 0.0010 at that level.
In a block experiment, the self-normalizing determinant retained transported-reference calibration. With a block-size parameter of 3, a sample size of 1,000 and a canonical correlation of 0.99, its simulated coverage was 0.949, close to the nominal 0.95. A nonnested product under the same setting covered only 0.836, while a plug-in interval restored coverage to 0.9492. The comparison shows why similar-looking denominator formulas can require different interval calculations.
The mediation simulation showed a different failure mode. With 1,000 observations, coverage stayed between 0.9493 and 0.9500 as the mediator residual variance moved from 1 to the setting equal to the sample size minus 2, but the median standard error rose from 0.0384 to 21.8537 effect units. With 100 observations, coverage ranged from 0.9373 to 0.9381. The pattern indicates a severe loss of precision without a corresponding change in the denominator law.
What the evidence separates
Tests with non-Gaussian sampling laws exposed two distinct calibration problems. Under exponential errors, the denominator’s actual relative variance was 40, compared with the Gaussian reference value of 10, and limiting denominator coverage was 0.6729, while effect Wald intervals were asymptotically valid. Under a dependent-error design, the denominator retained the Gaussian law, but Gaussian-reference effect-Wald coverage was 0.8731 at residual variance 1 and 0.7422 as the residual-variance sequence approached zero. Empirical standard errors corrected that latter failure.
The experiments separate denominator-law miscalibration from effect-variance miscalibration. They also separate both from the loss of precision seen when mediator residual variance collapses. The practical message is to examine denominator structure before choosing a reference law or confidence interval. Exact pivoting remains tied to fixed-dimensional Gaussian covariance sampling, while the non-Gaussian results rely on the specified sampling laws and empirical studentization.
Preprint status
The document is an arXiv version 2 preprint dated 24 September 2026. The author reports support from JSPS KAKENHI Grant Number 25K24203.
Paper data and sources
Original title: Self-Normalizing Denominators in Rational Causal Estimation
Authors: Shu Tamano
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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