Edgeworth++: Advanced Asymptotic Expansions

• Edgeworth++ is a refined framework that enhances classical Edgeworth expansions using cumulant- and Hermite-based corrections to improve Gaussian or semicircular approximations.
• It extends asymptotic theory to cover nonlinear, weighted, and self-normalized statistics, providing remainder control sufficient for second-order inference and robust local limit results.
• Applications include designing skewness-removing transforms (e.g., Fisher’s z), refining volatility models, and achieving precise approximations in free probability and combinatorial counting.

to=arxiv_search 天天中彩票未json {"^^^^^^^^7query7 expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^^^^^", "^^^^^^^^7max_results7 ^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7query7^^^^} to=arxiv_search RGCTXjson {"^^^^^^^^7query7 approximations for distributions of symmetric statistics\" OR ^^^^^^^^7ti:\7 transformation via Edgeworth expansion\" OR ^^^^^^^^7ti:\7 short Edgeworth expansions for weighted sums of random vectors\"", "^^^^^^^^7max_results7 ^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7query7^^^^} to=arxiv_search 大发时时彩怎么json {"^^^^^^^^7query7 expansions", "^^^^^^^^7max_results7 ^^^^7 OR ti:\7^^^^} As an Editor’s term, “Edgeworth++” denotes a family of refinements in which Edgeworth expansions are used not merely as a first correction to Gaussian asymptotics, but as a higher-order analytical device for nonlinear statistics, transformed statistics, weighted and self-normalized sums, stochastic-volatility functionals, Markov additive functionals, Wiener chaos, free probability, and combinatorial counting. The common pattern is explicit correction of a leading normal or semicircular approximation by cumulant- or Hermite-based terms together with remainder control strong enough for second-order inference, local limit theory, transport metrics, entropy, or structural classification of admissible models (&&&^^^^7query7^^^^&&&, &&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^&&&, &&&^^^^7query7^^^^&&&, Paul et al., 15 Oct 2025, &&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7query7^^^^&&&).

^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^. Baseline asymptotics and the Edgeworth principle

In the classical central limit setting, a normalized statistic PRESERVED_PLACEHOLDER_^^^^7query7^^^^ satisfies a Gaussian approximation of the form PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^. Edgeworth theory refines this by adding polynomial corrections multiplied by the Gaussian density, typically

PRESERVED_PLACEHOLDER_^^^^7max_results7^^^^

with a valid PRESERVED_PLACEHOLDER_^^^^7query7^^^^-term expansion having remainder PRESERVED_PLACEHOLDER_^^^^7ti:\7^^^^ when the relevant moment and smoothness conditions hold (&&&^^^^7query7^^^^&&&, &&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^&&&).

Within this perspective, “Edgeworth++” signifies a shift from first-order normal approximation and one-term correction to settings in which the expansion is used for stronger purposes: obtaining PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7^^^^ remainders for nonlinear symmetric statistics, designing skewness-removing transformations, constructing non-uniform bounds under fractional moments, or replacing the Gaussian reference by a signed Hermite-corrected measure adapted to a nonclassical limit regime (&&&^^^^7query7^^^^&&&, &&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7ti:\7^^^^&&&, Paul et al., 15 Oct 2025). This suggests that the label is best understood as an umbrella for advanced Edgeworth methodology rather than as the name of a single formal theorem.

^^^^7max_results7^^^^. High-precision asymptotics for nonlinear and asymptotically linear statistics

A central development concerns symmetric statistics that are not simple sums but admit a Hoeffding decomposition. For i.i.d. observations PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7^^^^, the statistic is decomposed as

$T-\mathbb{E}T = L+Q+K+R,$

with linear part $L=N^{-1/2}\sum_i g(X_i)$, quadratic part $Q=N^{-3/2}\sum_{i<j} v(X_i,X_j)$, cubic part PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7query7^^^^, and higher-order remainder PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^. Under Cramér’s condition on the linear projection, moment assumptions on PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7max_results7^^^^, difference-moment control of the remainder, and a non-reducibility condition on the quadratic kernel, the standardized distribution admits a two-term Edgeworth expansion with

PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7query7^^^^

which is precisely the scale needed for second-order efficiency comparisons such as Hodges–Lehmann deficiency (&&&^^^^7query7^^^^&&&).

The same “higher-order but still explicit” philosophy appears in randomized multivariate CLTs for weighted sums PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7ti:\7^^^^ of i.i.d. random vectors. Under PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7 OR ti:\7^^^^, PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7 OR ti:\7^^^^, and vanishing third mixed moments, the law of PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)77^^^^ is approximated by a signed measure with density

PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^8

where PRESERVED_PLACEHOLDER_^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^9 is an explicit quartic polynomial in the coordinates. Averaging over PRESERVED_PLACEHOLDER_^^^^7max_results7query7^^^^ uniformly on the sphere, the convex-set error is bounded by PRESERVED_PLACEHOLDER_^^^^7max_results7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^, extending Bobkov’s one-dimensional result to general dimension (&&&^^^^7max_results7^^^^&&&).

What distinguishes these results from textbook Edgeworth expansions is not only the order of the remainder but the structural breadth of the admissible statistics: PRESERVED_PLACEHOLDER_^^^^7max_results7max_results7^^^^-statistics of degree PRESERVED_PLACEHOLDER_^^^^7max_results7query7^^^^ or PRESERVED_PLACEHOLDER_^^^^7max_results7ti:\7^^^^, rank-based procedures, smooth empirical-measure functionals, and more general asymptotically linear estimators all fall under this program when the nonlinear part is sufficiently controlled (&&&^^^^7query7^^^^&&&).

^^^^7query7^^^^. Transformational use: skewness removal and variance stabilization

A particularly strong form of “Edgeworth++” uses the expansion not merely to approximate a fixed statistic, but to design the statistic itself. For the sample Pearson correlation PRESERVED_PLACEHOLDER_^^^^7max_results7 OR ti:\7^^^^ from a bivariate normal sample, Edgeworth coefficients for a general transform PRESERVED_PLACEHOLDER_^^^^7max_results7 OR ti:\7^^^^ can be computed symbolically from the joint central MGF of five sample means. Requiring the skewness term PRESERVED_PLACEHOLDER_^^^^7max_results77^^^^ to vanish leads to the differential equation

PRESERVED_PLACEHOLDER_^^^^7max_results7^^^^8

whose nontrivial solutions are PRESERVED_PLACEHOLDER_^^^^7max_results7^^^^9. Thus Fisher’s PRESERVED_PLACEHOLDER_^^^^7query7query7^^^^ emerges as the unique skewness-eliminating transformation at the relevant order (&&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^&&&).

The same calculation yields a refined characterization of the transformed statistic: PRESERVED_PLACEHOLDER_^^^^7query7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^ The resulting Edgeworth density has PRESERVED_PLACEHOLDER_^^^^7query7max_results7^^^^ relative accuracy, and in the example PRESERVED_PLACEHOLDER_^^^^7query7query7^^^^ the maximum error of the induced density approximation is about PRESERVED_PLACEHOLDER_^^^^7query7ti:\7^^^^, while the basic Fisher approximation exceeds PRESERVED_PLACEHOLDER_^^^^7query7 OR ti:\7^^^^ (&&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^&&&).

This transformation-based viewpoint generalizes. The same paper formulates a symbolic recipe for any statistic expressible as a smooth function of several sample means: compute low-order moments from the multivariate MGF, derive PRESERVED_PLACEHOLDER_^^^^7query7 OR ti:\7^^^^, solve for transforms that eliminate skewness, and then apply the Edgeworth density. In this sense, “Edgeworth++” means using cumulant structure as a design principle for variance-stabilizing and symmetry-improving transformations rather than treating higher-order asymptotics as a purely descriptive afterthought (&&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^&&&).

^^^^7ti:\7^^^^. Dependence, self-normalization, and non-uniform local theory

A major extension of Edgeworth theory replaces uniform approximation by weighted, non-uniform control. For normalized sums of i.i.d. variables with only fractional moment PRESERVED_PLACEHOLDER_^^^^7query77^^^^, PRESERVED_PLACEHOLDER_^^^^7query7^^^^8, one can still define a truncated Edgeworth density PRESERVED_PLACEHOLDER_^^^^7query7^^^^9 with PRESERVED_PLACEHOLDER_^^^^7ti:\7query7^^^^ and prove

PRESERVED_PLACEHOLDER_^^^^7ti:\7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^

together with a sharper bound weighted by PRESERVED_PLACEHOLDER_^^^^7ti:\7max_results7^^^^. This extends Petrov-type non-uniform local limit theorems to non-integer moment orders and makes the tail behavior part of the asymptotic statement itself (&&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7ti:\7^^^^&&&).

For self-normalized sums PRESERVED_PLACEHOLDER_^^^^7ti:\7query7^^^^, the same philosophy becomes substantially ավելի technical because the characteristic function no longer factorizes. Conditioning on PRESERVED_PLACEHOLDER_^^^^7ti:\7ti:\7^^^^ restores a product structure and produces conditional Edgeworth polynomials PRESERVED_PLACEHOLDER_^^^^7ti:\7 OR ti:\7^^^^; averaging yields an unconditional expansion

PRESERVED_PLACEHOLDER_^^^^7ti:\7 OR ti:\7^^^^

with PRESERVED_PLACEHOLDER_^^^^7ti:\77^^^^ for odd PRESERVED_PLACEHOLDER_^^^^7ti:\7^^^^8 under symmetry. The non-uniform CLT bound takes the form

PRESERVED_PLACEHOLDER_^^^^7ti:\7^^^^9

and the local theory yields density expansions, an entropic CLT, and a total variation CLT for PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7query7^^^^ under minimal moment hypotheses (&&&^^^^7query7^^^^&&&).

For integer-valued additive functionals of uniformly elliptic inhomogeneous Markov chains, the local expansion acquires an explicitly arithmetic form. Instead of a purely Gaussian-Hermite correction, one obtains

PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^

so the asymptotics are Gaussian times polynomials times trigonometric polynomials. The standard Edgeworth expansion of order PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7max_results7^^^^ holds in a conditionally stable way if and only if conditional distributions of PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7query7^^^^ are PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7ti:\7^^^^-close to uniform, uniformly in the conditioning coordinates (&&&^^^^7 OR ti:\7^^^^&&&).

A related but metric-oriented extension treats discrete-time volatility models through Bernoulli-shift dependence coefficients. Under summability of physical dependence coefficients and a non-lattice small-ball condition, first-order Edgeworth approximations are obtained in Kolmogorov distance, and a Gamma-corrected Gaussian proxy yields PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7 OR ti:\7^^^^-error PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7 OR ti:\7^^^^ for Hölder-type functions of augmented GARCHPRESERVED_PLACEHOLDER_^^^^7 OR ti:\77^^^^, iterated random functions, linear processes, and Volterra processes (&&&^^^^7ti:\7^^^^&&&).

^^^^7 OR ti:\7^^^^. Nonclassical limit objects: Wiener chaos and free probability

In Wiener chaos, the small parameter is no longer PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7^^^^8 but the variance of the carré-du-champ. For PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7^^^^9 with PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7query7^^^^ and PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^, the order-PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7max_results7^^^^ Edgeworth-type signed measure has density

PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7query7^^^^

The total variation error is bounded by

PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7ti:\7^^^^

and PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7 OR ti:\7^^^^ is comparable to the fourth cumulant. This yields arbitrary-order Edgeworth expansions on fixed Wiener chaoses and recovers the optimal fourth moment theorem as a byproduct (Paul et al., 15 Oct 2025).

In free probability, the role of the Gaussian law is played by the semicircle law. For normalized free sums PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7 OR ti:\7^^^^ with PRESERVED_PLACEHOLDER_^^^^7 OR ti:\77^^^^, PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7^^^^8, and PRESERVED_PLACEHOLDER_^^^^7 OR ti:\7^^^^9, the density $T-\mathbb{E}T = L+Q+K+R,$^^^^7query7^^^^ is approximated by a Meixner-type proxy

$T-\mathbb{E}T = L+Q+K+R,$^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^

where $T-\mathbb{E}T = L+Q+K+R,$^^^^7max_results7^^^^ are explicit functions of $T-\mathbb{E}T = L+Q+K+R,$^^^^7query7^^^^ and $T-\mathbb{E}T = L+Q+K+R,$^^^^7ti:\7^^^^. This leads not only to local density expansions but also to entropic and free Fisher-information asymptotics such as

$T-\mathbb{E}T = L+Q+K+R,$^^^^7 OR ti:\7^^^^

with $T-\mathbb{E}T = L+Q+K+R,$^^^^7 OR ti:\7^^^^ remainders when higher moments are available (&&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7query7^^^^&&&).

These two lines show that “Edgeworth++” is not tied to classical triangular arrays. The same Hermite-correction logic survives when the approximation target is a signed Gaussian perturbation on Wiener space or a semicircle-based Meixner proxy in free convolution asymptotics (Paul et al., 15 Oct 2025, &&&^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7query7^^^^&&&).

^^^^7 OR ti:\7^^^^. Applied uses, structural criteria, and terminological scope

Several applications use Edgeworth corrections as operational tools rather than purely asymptotic descriptors. In spot-volatility estimation for a continuous Itô semimartingale with drift and leverage, the studentized statistic $T-\mathbb{E}T = L+Q+K+R,$7 admits an Edgeworth expansion

$T-\mathbb{E}T = L+Q+K+R,$8

with explicit Hermite-polynomial corrections. The induced confidence intervals improve coverage accuracy from $T-\mathbb{E}T = L+Q+K+R,$9 to $L=N^{-1/2}\sum_i g(X_i)$^^^^7query7^^^^ in the one-sided case and from $L=N^{-1/2}\sum_i g(X_i)$^^^^7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^ to $L=N^{-1/2}\sum_i g(X_i)$^^^^7max_results7^^^^ in the two-sided case (&&&^^^^7query7query7^^^^&&&).

In stochastic-volatility option pricing, Edgeworth expansion enters as a rigorous correction around the Black–Scholes price. For a general ergodic diffusion volatility model, the European payoff satisfies

$L=N^{-1/2}\sum_i g(X_i)$^^^^7query7^^^^

uniformly over bounded payoff functions. This validates singular perturbation formulas for fast mean-reverting stochastic volatility by identifying the first Edgeworth correction to the Black–Scholes law (&&&^^^^7query7Edgeworth expansions arXiv (Götze et al., 2021, Vrbik, 2022, Ayvazyan, 2023, Beckedorf et al., 2022, Jirak, 2021, Dolgopyat et al., 2022)7^^^^&&&).

A different application converts counting problems into local probability asymptotics. For integer points in a polytope defined by separable constraints, maximum-entropy tilting gives an exact identity

$L=N^{-1/2}\sum_i g(X_i)$^^^^7ti:\7^^^^

and $L=N^{-1/2}\sum_i g(X_i)$^^^^7 OR ti:\7^^^^ is approximated by a Gaussian density times the Edgeworth factor

$L=N^{-1/2}\sum_i g(X_i)$^^^^7 OR ti:\7^^^^

The framework is applied to contingency tables with fixed margins and to graphs with a given degree sequence, where the correction is asymptotically valid in growing dimension (&&&^^^^7query7max_results7^^^^&&&).

Across these settings, a recurring obstacle is arithmetic or geometric pathology in the nonleading terms. The literature addresses it in several distinct ways: Cramér’s condition on the linear projection and non-reducibility of the quadratic kernel for symmetric statistics (&&&^^^^7query7^^^^&&&); vanishing third mixed moments for weighted sums of random vectors (&&&^^^^7max_results7^^^^&&&); small-ball and non-lattice conditions for volatility models (&&&^^^^7ti:\7^^^^&&&); and conditional mod-$L=N^{-1/2}\sum_i g(X_i)$7 equidistribution for integer-valued Markov additive functionals (&&&^^^^7 OR ti:\7^^^^&&&). This suggests that “Edgeworth++” is less a single theorem than a strategy for identifying the precise regularity needed to make higher-order corrections survive the specific dependence, lattice, or nonlinear structure of the problem.

The label should also be distinguished from uses of “Edgeworth” that are unrelated to Edgeworth expansions. In “The Edgeworth Conjecture with Small Coalitions and Approximate Equilibria in Large Economies” the name refers to Francis Ysidro Edgeworth’s conjecture in general equilibrium rather than to asymptotic expansion theory (&&&^^^^7query77^^^^&&&). Likewise, “Dust in the Edgeworth-Kuiper belt zone” concerns orbital dynamics in the Edgeworth–Kuiper belt and not probabilistic asymptotics (&&&^^^^7query7^^^^8&&&).