Non-Uniform Edgeworth Expansions

• Non-uniform Edgeworth expansions are asymptotic series that refine Gaussian approximations by incorporating correction terms with precise tail decay.
• They provide explicit error bounds of the form C Bₙ^(–(r+1)) (1+|x|)^(–m), offering improved accuracy in the tails compared to uniform bounds.
• The methodology uses characteristic function analysis, cumulant control, and spectral techniques to extend results to dependent, non-i.i.d., and self-normalized statistics.

Non-uniform Edgeworth expansions are asymptotic series that refine the normal approximation to sums (or functionals) of random variables, providing explicit correction terms whose accuracy can be sharply tracked as a function of both the sample size and the point at which the approximation is evaluated. The non-uniformity refers to remainder estimates that decay faster as one moves away from the center, typically featuring weights of the form $(1+|x|)^{-m}$. This stands in contrast to classical (uniform) expansions, which bound the global sup-norm of the error. The modern theory rigorously establishes non-uniform Edgeworth expansions for a broad class of dependent and inhomogeneous sequences, Markov chains, dynamical systems, products of random matrices, non-i.i.d. arrays, and even self-normalized or weighted statistics, using tools of spectral analysis, analytic perturbation, characteristic function techniques, and cumulant control.

1. General Formulation and Basic Principles

Consider a sequence or triangular array of random variables $X_1, X_2, \dots, X_n$ and their normalized sum

$$W_n = \frac{S_n - A_n}{B_n}, \qquad S_n = \sum_{k=1}^n X_k, \quad B_n > 0, \quad A_n = O(B_n^{-1}).$$

The distribution function under study is $F_n(x) = \mathbb{P}(W_n \leq x)$. The Edgeworth expansion provides an asymptotic series:

$$F_n(x) = \Phi(x) + \sum_{j=1}^r B_n^{-j} H_{j,n}(x)\varphi(x) + R_{n,r}(x),$$

where $\Phi$ and $\varphi$ are the standard normal cdf and pdf, $H_{j,n}$ are Edgeworth polynomials (in general depending on $n$ and $x$), and $R_{n,r}(x)$ is a remainder term. Non-uniformity enters through bounds of the form

$$|F_n(x) - F_{n,r}(x)| \leq C B_n^{-(r+1)} (1+|x|)^{-m},$$

for some integer $m \geq r+2$, providing polynomial decay in the tails. When $B_n \sim n^{1/2}$ and $r$ is fixed, this ensures strong tail control and precise rates at all $x \in \mathbb{R}$ (Hafouta, 2022, Hafouta, 9 Nov 2025).

The Edgeworth polynomials $H_{j,n}(x)$ (or $P_j(x)$ in some conventions) are universal combinatorial expressions in the normalized cumulants or moments up to order $m+1$, and can be written as

$$P_p(x) = \sum_{k: \sum \ell k_\ell = p} C_k \prod_{\ell} \kappa_{\ell+2}^{k_\ell} H_{p-1}(x),$$

where the constants $C_k$ arise from the combinatorial expansion of the limiting log-characteristic function (Hafouta, 2022, Bobkov et al., 2011).

2. Hypotheses and Analytical Conditions

Sharp non-uniform Edgeworth expansions rely on two classes of analytic hypotheses:

• (A) Local Control ("small-$t$" regime): Bounding higher derivatives of the log-characteristic function,

$$\phi_n(t) = \mathbb{E} e^{itW_n}, \quad A_n(t) = \log \phi_n(t) + t^2/2,$$

there exists $m \geq 3$ such that for $|t| \leq \epsilon B_n$,

$$|A_n^{(j)}(t)| \leq C_j B_n^{-(j-2)}, \quad \forall\; 3 \leq j \leq m+1.$$

• (B) Cumulant-Growth/Weak Stationarity and Tail Control ("large-$t$" regime): Ensuring cumulants $\kappa_k(S_n)=n\kappa_k+O(\rho^n)$ for $k = 2,\ldots,m+1$, $0<\rho<1$, so that cumulants exist and grow linearly with $n$. For the tails, integrals of the characteristic function satisfy, for large $B$,

$$\int_{c B_n}^{B B_n^{(m-3)/2}} |\phi_n^{(m)}(t)||t|^{-1} dt = o(B_n^{-(m-2)}).$$

In applications, these are typically established through spectral gap and analytic perturbation theory for suitably defined transfer operators on Banach spaces (Hafouta, 2022, Hafouta, 9 Nov 2025).

The order of the expansion $r$ (highest $B_n^{-j}$ term) is constrained by the available moments: $1 \leq r \leq m-2$, and $m$ is typically taken as the greatest integer less than the available (possibly fractional) moment order (Bobkov et al., 2011, Beckedorf et al., 2022).

3. Main Results: Expansion Statements and Remainder Bounds

The primary theorem can be formulated as follows: For $1 \leq r \leq m-2$,

$$F_n(x) = \Phi(x) + \sum_{j=1}^r B_n^{-j} H_{j,n}(x)\varphi(x) + R_{n,r}(x),$$

with the sharp remainder estimate,

$|R_{n,r}(x)| \leq C B_n^{-(r+1)} (1+|x|)^{-m},$

where $C$ depends only on moments and analytic constants up to order $m+1$ (Hafouta, 2022, Hafouta, 9 Nov 2025, Bobkov et al., 2011).

For i.i.d. summands with $E|X_1|^s<\infty$ ($s \geq 2$ possibly non-integer, $m = [s]$), the non-uniform density approximation (local limit) is

$$P_n(x) = \varphi(x) \left[ 1 + \sum_{k=1}^{m-2} n^{-k/2} P_k(x) \right] + R_n(x),$$

and for $s=m$ integer, the remainder is $O(n^{-(m-2)/2} (1+|x|)^{-m})$; for fractional moments $2$O(n^{-(s-2)/2}(1+|x|^s))$. Here, $P_k(x)$ are Hermite-type polynomials determined by the cumulants of $X_1$ up to order $m+2$ (Bobkov et al., 2011).

For general weakly dependent or non-i.i.d. cases—including weighted arrays, Markov chains, dynamical systems, and functionals of weighted empirical distributions—analogous expansions hold using generalizations of von Mises calculus to signed measures (Hafouta, 9 Nov 2025, Withers et al., 2010).

A key feature is the explicit decay rate of the remainder with $x$, improving substantially upon uniform error bounds:

$$\sup_{x \in \mathbb{R}} (1+|x|)^m |F_n(x) - \Phi(x)| \leq C n^{-1/2},$$

illustrating that the Gaussian approximation's error is polynomially small even for large deviations, provided enough moments exist (Hafouta, 2022).

4. Methodology: Construction and Proof Strategies

The analytical core involves characteristic function analysis and Fourier inversion, with key steps as follows:

• Taylor Expansion and Cumulant Control: Write the log-characteristic function as a Taylor series at $t=0$, matching terms with cumulants up to degree $m+1$ (Hafouta, 2022, Bobkov et al., 2011).
• Splitting Small and Large $t$: For $|t| \leq cB_n$, use the expansion and analytic bounds; for large $t$, decay estimates from moment control and analytic perturbation yield exponentially small contributions. Esseen-type smoothing inequalities are used to balance these regimes optimally (Hafouta, 9 Nov 2025).
• Hermite Polynomial Representation: Edgeworth polynomials are constructed to ensure the term-by-term cancellation of the error to each desired order, with explicit combinatorial formulas involving Hermite polynomials and cumulants (Hafouta, 2022, Bobkov et al., 2011, Withers et al., 2010).
• Control of the Remainder: For each $m$ available moment, the non-uniformity in $x$ is achieved by repeated integration by parts and careful tracking of the way polynomial weights in $x$ emerge from the Fourier inversion. For dependent structures, analytic perturbation of the transfer operator supplies the necessary regularity (Hafouta, 2022, Hafouta, 9 Nov 2025).

5. Classes of Examples and Applications

Non-uniform Edgeworth expansions have been established for a wide array of settings, including but not limited to:

Class of Processes	Mechanism for Assumption Verification	Notable Features
I.i.d. or non-i.i.d. sums	Moment conditions, characteristic function	Petrov–Götze bounds, fractional moments, local moderate deviation rates (Bobkov et al., 2011)
Inhomogeneous Markov chains	Spectral gap, transfer operator quasi-compactness	Stationary expansions, slow variance growth (Hafouta, 2022, Hafouta, 9 Nov 2025)
Products of random matrices	Projective operator, Furstenberg–Kesten theory	All orders, exponential moment (Hafouta, 2022, Hafouta, 9 Nov 2025)
(Partially) expanding/hyperbolic dynamics	Ruelle–Perron–Frobenius operator, Young towers	Exponential tails, stationary and non-stationary cases (Hafouta, 2022, Hafouta, 9 Nov 2025)
Weighted empirical measures / functionals	von Mises calculus for signed measures	Cumulant expansions for smooth functionals, Cornish–Fisher quantiles (Withers et al., 2010)
Self-normalized statistics (e.g., Student $T$)	Conditioning/deconvolution, non-normalized CFs	Edgeworth expansions for $S_n/V_n$ with strong tail accuracy (Beckedorf et al., 2022)

Key applications include:

• Edgeworth expansions for moments and smooth functionals $h(W_n)$ with asymptotic series for $\mathbb{E}[h(W_n)]$ where $|h'(x)|(1+|x|)^m$ is integrable,
• Strong Berry–Esseen theorems and Wasserstein metric expansions: $W_p(\mathrm{Law}(W_n), N(0,1)) = O(B_n^{-1})$, with higher-order expansions in $W_p$ (Hafouta, 9 Nov 2025),
• Enhanced local large deviation and moderate deviation results due to tail-accuracy,
• Improved entropic central limit theorems and total variation rates for self-normalized sums (Beckedorf et al., 2022),
• Explicit asymptotic corrections for quantiles in nonparametric statistics via Cornish–Fisher expansions (Withers et al., 2010).

6. Optimality, Range of Validity, and Comparison with Uniform Expansions

The attainable order $r$ in the expansion is sharply determined by the maximal integer $m$ such that moments up to order $m+1$ (or cumulants up to $m+1$) can be controlled. The exponent in the non-uniform polynomial weight can, in regular examples, be taken arbitrarily large provided enough moments exist. For fractional moments $s>2$, the degree of non-uniformity interpolates continuously between the classical Petrov regime (integer $m$) and the lower-moment case (Bobkov et al., 2011).

In stationary or weakly stationary scenarios—in particular, for Markov chains and dynamical systems with transfer operator spectral gaps—the Edgeworth polynomials do not accumulate $n$-dependence, and one recovers "stationary" polynomials as in the classical independent case (Hafouta, 2022, Hafouta, 9 Nov 2025).

The non-uniform expansions are strictly stronger than uniform expansions:

• Uniform expansions control $\sup_{x}|F_n(x)-\Phi(x)|$ at a slower rate and do not capture the rapid decay of the error in the tails.
• Non-uniform expansions serve as the essential technical input for fine-grained Berry–Esseen results, $L^p$-approximations, and optimal transport error rates.

The results are valid under minimal regularity, requiring only moment conditions and analytic control available from spectral gap or quasi-compactness methods, operator-analytic perturbations, or, for weighted and non-i.i.d. arrays, from suitably bounded von Mises derivatives (Hafouta, 2022, Withers et al., 2010).

7. Limitations, Extensions, and Modern Directions

The necessity of controlling cumulants and moments up to order $m+1$ constrains the maximal order of expansion and the polynomial decay exponent. In certain pathological cases, such as lacking spectral gaps or exponential tails, the expansions may not be achievable to arbitrary order. However, the methods extend beyond independence and stationary settings—encompassing weak dependence, non-identically distributed arrays, sequential or random dynamical systems, and broad classes of functionals, indicating the robustness of the approach (Hafouta, 9 Nov 2025, Hafouta, 2022, Withers et al., 2010). Applications to self-normalized processes, entropic CLTs, and Wasserstein metrics continue to expand the relevance of non-uniform Edgeworth asymptotics, with explicit formulas for moments, quantiles, and functionals becoming computationally feasible under the explicit bracket-and-cumulant calculus established in the referenced papers.