Conditional Cramér–Edgeworth Expansion

• Conditional Cramér–Edgeworth expansion is a methodological framework that approximates conditional distributions via tilted measures, cumulant expansions, and Hermite corrections.
• It is applied across diverse settings—extreme-value regimes, Markov chains, urn models, and shot noise—with model-specific remainder controls ensuring precise asymptotic results.
• The approach quantifies thresholds for conditional independence and addresses arithmetic obstructions, supporting applications in large deviations, metastability, and rare-event simulations.

Search arXiv for "conditional Edgeworth expansion Cramér conditional Gibbs principle" Conditional Cramér–Edgeworth expansion denotes a family of asymptotic representations for conditional laws, conditional tail probabilities, or conditionally stabilized local limits in which the leading term is either Gaussian or exponentially tilted, and higher-order corrections are written through cumulants and Hermite-polynomial terms. In the cited literature, conditioning appears in several distinct forms: on extreme exceedances of a sum, on the exact value of a normalized sum in a super-large deviation regime, on a finite collection of past states for additive functionals of uniformly elliptic Markov chains, on a sum constraint in generalized urn models, and on a fixed shot-noise total. Across these settings, the expansion is obtained by a change of measure, Fourier inversion, and model-specific control of the remainder; the resulting formulas quantify when conditioning preserves Gaussian-type asymptotics, when it produces oscillatory lattice corrections, and when it destroys asymptotic independence (Biret et al., 2016, Dolgopyat et al., 2022, Fan et al., 2012, Mirakhmedov et al., 2014, Járai, 2019, Cao, 2013).

1. Principal settings and conditional objects

The topic does not refer to a single theorem, but to a recurring asymptotic mechanism. In the i.i.d. extreme-deviation setting, one studies the conditional density of one coordinate or of a fixed block given the rare event $\{S_n>n a_n\}$, where $S_n=X_1+\cdots+X_n$ and $a_n\to\infty$ (Biret et al., 2016). In the super-large deviation setting of Cao, the focus is the local behavior of $S_n$ at $n a_n$, derived from the $n$-fold convolution of a normalized tilted density under a triangular array (Cao, 2013). In generalized urn models, the random frequency vector is represented as conditionally independent given its sum, and Edgeworth expansions are derived for decomposable statistics under that conditioning (Mirakhmedov et al., 2014).

In lattice Markov-chain problems, the conditional object is different: one expands $\mathbb P\{S_N=k\}$ and its conditioned variants given $X_{j_1},\dots,X_{j_\ell}$, with the possibility of additional oscillatory factors $e^{2\pi i a k/J}$ coming from arithmetic resonances (Dolgopyat et al., 2022). In martingale large deviations, the conditioning is encoded through the filtration and the conjugate distribution technique of Cramér; the resulting expansion is expressed through conditional cumulants and conditional Bernstein bounds rather than through an event of the form $\{S_n=n a_n\}$ (Fan et al., 2012). In Poisson shot noise, the expansion is used to approximate the conditional density of $S_n=X_1+\cdots+X_n$0 given $S_n=X_1+\cdots+X_n$1, uniformly in $S_n=X_1+\cdots+X_n$2 (Járai, 2019).

Setting	Conditional object	Leading form
Extreme i.i.d. sums	$S_n=X_1+\cdots+X_n$3, or $S_n=X_1+\cdots+X_n$4	Tilted density $S_n=X_1+\cdots+X_n$5 with Hermite corrections
Lattice additive functionals	$S_n=X_1+\cdots+X_n$6	Gaussian term times polynomials and trigonometric factors
Sum-constrained models	$S_n=X_1+\cdots+X_n$7, or $S_n=X_1+\cdots+X_n$8	Gaussian or tilted term with cumulant corrections

A plausible implication is that “conditional Cramér–Edgeworth expansion” is best understood as a methodological class rather than a single canonical formula.

2. Conditioning by tilting, conjugate measures, and operator perturbation

The standard i.i.d. construction begins with a tilted density

$S_n=X_1+\cdots+X_n$9

where $a_n\to\infty$0, $a_n\to\infty$1, $a_n\to\infty$2, and $a_n\to\infty$3 for $a_n\to\infty$4. For a threshold sequence $a_n\to\infty$5, one chooses $a_n\to\infty$6 so that $a_n\to\infty$7, and then uses $a_n\to\infty$8 as the reference law for the conditioned coordinate (Biret et al., 2016). Cao’s super-large deviation analysis uses the same tilt in the form

$a_n\to\infty$9

with $S_n$0 defined by $S_n$1, and studies the triangular array obtained by normalizing i.i.d. variables under $S_n$2 (Cao, 2013).

For martingales, the analogue of the tilted law is the conjugate measure $S_n$3, defined through the martingale

$S_n$4

Under $S_n$5, the sum decomposes as

$S_n$6

with $S_n$7, $S_n$8, and cumulant-generating process $S_n$9 (Fan et al., 2012). The saddle-point parameter is then chosen by solving $n a_n$0.

In the shot-noise problem, the same logic is expressed through a Cramér transform. For

$n a_n$1

one introduces a parameter $n a_n$2 so that under the tilted measure $n a_n$3, and rewrites the density through a contour-shifted Fourier inversion formula (Járai, 2019). In the lattice Markov setting, the role of tilting near nonzero resonant points is played by a sequential Perron–Frobenius perturbation theorem giving analyticity of the transfer-operator’s top eigenvalue $n a_n$4 and conditional cumulant-generating functions on blocks $n a_n$5 (Dolgopyat et al., 2022).

These constructions are formally different, but they all replace the original conditioned problem by an auxiliary law under which cumulants can be expanded explicitly.

3. Canonical algebraic forms

In the extreme i.i.d. exceedance problem, setting

$n a_n$6

one obtains, for any fixed integer $n a_n$7,

$n a_n$8

uniformly in $n a_n$9. Each $n$0 is an explicit linear combination of Hermite polynomials with coefficients that are universal polynomials in the tilted cumulants. The first two corrections are

$n$1

with $n$2 and $n$3 (Biret et al., 2016).

Cao’s triangular-array result has the same Hermite structure. If

$n$4

and $n$5 denotes the normalized density of the $n$6-fold convolution, then uniformly in $n$7,

$n$8

After translating back to the original sum, this yields a local limit at $n$9 with an exponential factor $\mathbb P\{S_N=k\}$0, the scale $\mathbb P\{S_N=k\}$1, and higher-order corrections involving tilted cumulants $\mathbb P\{S_N=k\}$2 (Cao, 2013).

In generalized urn models, decomposable statistics satisfy

$\mathbb P\{S_N=k\}$3

where $\mathbb P\{S_N=k\}$4 has degree $\mathbb P\{S_N=k\}$5 and is built from joint cumulants of the pairs $\mathbb P\{S_N=k\}$6. The first two polynomial corrections again have the standard Hermite pattern: $\mathbb P\{S_N=k\}$7 (Mirakhmedov et al., 2014).

The Markov-chain lattice expansion differs in one decisive respect. There is an integer $\mathbb P\{S_N=k\}$8 and polynomial families $\mathbb P\{S_N=k\}$9 such that

$X_{j_1},\dots,X_{j_\ell}$0

uniformly in $X_{j_1},\dots,X_{j_\ell}$1. Under the paper’s non-concentration-mod-$X_{j_1},\dots,X_{j_\ell}$2 condition, the terms with $X_{j_1},\dots,X_{j_\ell}$3 vanish and one recovers the usual single-term lattice Edgeworth expansion (Dolgopyat et al., 2022).

For martingales, the expansion is presented in large-deviation form: $X_{j_1},\dots,X_{j_\ell}$4 with first coefficients

$X_{j_1},\dots,X_{j_\ell}$5

and

$X_{j_1},\dots,X_{j_\ell}$6

where $X_{j_1},\dots,X_{j_\ell}$7 are conditional cumulants (Fan et al., 2012).

4. Structural assumptions and remainder control

The extreme-deviation i.i.d. theory assumes a density

$X_{j_1},\dots,X_{j_\ell}$8

with $X_{j_1},\dots,X_{j_\ell}$9 convex, four-times differentiable, and growing fast enough at infinity; equivalently, $e^{2\pi i a k/J}$0 lies in the class $e^{2\pi i a k/J}$1 of regularly- or rapidly-varying functions. The perturbation $e^{2\pi i a k/J}$2 is bounded and of smaller-order variation. Under these conditions, a sharp Abelian theorem yields

$e^{2\pi i a k/J}$3

The final remainder estimate takes the form

$e^{2\pi i a k/J}$4

so the expansion is uniformly valid on the full real line in the $e^{2\pi i a k/J}$5-variable (Biret et al., 2016).

Cao’s Abelian theorem imposes the Cramér condition on the moment-generating function, steepness on $e^{2\pi i a k/J}$6, and the requirement $e^{2\pi i a k/J}$7. It then gives

$e^{2\pi i a k/J}$8

so the skewness of the tilted law vanishes in the super-large deviation regime (Cao, 2013).

In the martingale setting, the assumptions are conditional Bernstein’s condition and variance regularity: $e^{2\pi i a k/J}$9 together with $\{S_n=n a_n\}$0, where $\{S_n=n a_n\}$1. For every fixed $\{S_n=n a_n\}$2, the remainder satisfies, uniformly in $\{S_n=n a_n\}$3,

$\{S_n=n a_n\}$4

(Fan et al., 2012).

The generalized urn expansion is driven by a Lindeberg-type condition $\{S_n=n a_n\}$5, moment assumptions up to order $\{S_n=n a_n\}$6, and smoothing conditions on characteristic functions. Its remainder satisfies

$\{S_n=n a_n\}$7

and under extra smoothing the range may be extended to $\{S_n=n a_n\}$8 (Mirakhmedov et al., 2014).

For power-law shot noise, writing $\{S_n=n a_n\}$9, Theorem 1 gives

$S_n=X_1+\cdots+X_n$00

whenever $S_n=X_1+\cdots+X_n$01, uniformly in the tilt parameter $S_n=X_1+\cdots+X_n$02 (Járai, 2019). This suggests that explicit remainder control is one of the defining features of the conditional theory, but the control mechanism is model-dependent.

5. Independence thresholds, arithmetic obstructions, and conditional stability

One of the central results in the extreme-deviation Gibbs framework is that conditional asymptotic independence is not automatic. Fix $S_n=X_1+\cdots+X_n$03. In the moderate-growth regime

$S_n=X_1+\cdots+X_n$04

the joint conditional density of $S_n=X_1+\cdots+X_n$05 given $S_n=X_1+\cdots+X_n$06 satisfies

$S_n=X_1+\cdots+X_n$07

so the first $S_n=X_1+\cdots+X_n$08 coordinates become asymptotically independent. In the fast-growth regime,

$S_n=X_1+\cdots+X_n$09

the second-order correction in the joint expansion no longer factorizes, and finite-block independence fails. The independence threshold is exactly

$S_n=X_1+\cdots+X_n$10

(Biret et al., 2016).

The lattice Markov-chain theory identifies a different obstruction: arithmetic resonance. Standard Edgeworth expansions of order $S_n=X_1+\cdots+X_n$11 hold in a conditionally stable way if and only if, for every $S_n=X_1+\cdots+X_n$12 and every $S_n=X_1+\cdots+X_n$13, the conditional distribution of $S_n=X_1+\cdots+X_n$14 given $S_n=X_1+\cdots+X_n$15 mod $S_n=X_1+\cdots+X_n$16 is $S_n=X_1+\cdots+X_n$17 close to uniform, uniformly in the choice of conditioning indices. Equivalently,

$S_n=X_1+\cdots+X_n$18

In particular, the local-limit theorem ($S_n=X_1+\cdots+X_n$19) is stable under such conditioning if and only if the conditional modulo-$S_n=X_1+\cdots+X_n$20 law converges to uniform at rate $S_n=X_1+\cdots+X_n$21 (Dolgopyat et al., 2022).

Generalized urn models provide a third perspective: the random vector of frequencies is viewed as conditionally independent given the sum, and the Edgeworth expansion exploits that representation (Mirakhmedov et al., 2014). The literature therefore explicitly separates three phenomena that are often conflated: conditional independence by representation, asymptotic independence under rare-event conditioning, and conditional stability of an Edgeworth expansion after conditioning on part of the process. These are not equivalent.

6. Applications and significance

In generalized urn models, the conditional expansion yields explicit asymptotics for decomposable statistics in sampling with replacement, sampling without replacement, and the multicolor Pólya–Egenberger scheme. The examples treated include the chi-square statistic, the sample sum in the without-replacement scheme, and Dixon’s spacing-frequency statistic for two-sample homogeneity tests (Mirakhmedov et al., 2014). In the chi-square example, the paper gives a two-term Edgeworth expansion; in the without-replacement example, it recovers the familiar first correction of order $S_n=X_1+\cdots+X_n$22.

For uniformly elliptic Markov chains, the expansions support a quantitative Prokhorov-type theorem, a conditionally stable local-limit theorem, and recovery of the “super-stable” Edgeworth theorems of Dolgopyat–Hafouta in the case of bounded independent summands. The paper also states that these formulas can be used for explicit pre-asymptotic error bounds in metastability, rare-event simulation, and statistical-mechanics models with integer observables (Dolgopyat et al., 2022).

In power-law shot noise, the Edgeworth expansion is turned into an approximation scheme for the conditional density $S_n=X_1+\cdots+X_n$23 to any desired accuracy, with error bounds uniform in $S_n=X_1+\cdots+X_n$24. The same work proves a stochastic comparison between $S_n=X_1+\cdots+X_n$25 given $S_n=X_1+\cdots+X_n$26 and unconditioned radii $S_n=X_1+\cdots+X_n$27, and gives an algorithmic scheme in which the overall error is $S_n=X_1+\cdots+X_n$28, with $S_n=X_1+\cdots+X_n$29 and polynomial growth in $S_n=X_1+\cdots+X_n$30 (Járai, 2019).

In the extreme-sum and super-large deviation literature, the expansion is tied directly to the conditional Gibbs principle. Cao develops an Abelian theorem and a triangular-array Edgeworth expansion for the regime $S_n=X_1+\cdots+X_n$31, while Biret–Broniatowski–Cao study the conditional distribution under $S_n=X_1+\cdots+X_n$32 and identify the threshold at which finite-block asymptotic independence emerges or fails (Cao, 2013, Biret et al., 2016).

Taken together, these results place conditional Cramér–Edgeworth expansion at the intersection of large deviations, local limit theory, and conditional limit theorems. The common structure is a cumulant expansion under a transformed measure; the decisive differences are the conditioning mechanism, the presence or absence of arithmetic resonances, and the regime in which the conditional law does or does not decouple.