Tweedie Calculus

Santiago Torres

arxiv: 2604.14486 · v2 · submitted 2026-04-15 · 🧮 math.ST · econ.EM· stat.ME· stat.TH

Tweedie Calculus

Santiago Torres This is my paper

Pith reviewed 2026-05-10 11:24 UTC · model grok-4.3

classification 🧮 math.ST econ.EMstat.MEstat.TH

keywords Tweedie representationsadditive noise modelsconditional expectationtempered distributionsFourier transformempirical Bayesdifferential privacy

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The pith

In additive noise models, conditional expectations of latent variables are given directly by a unique continuous linear functional applied to the observed density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general theory showing when posterior expectations of unobserved latent variables can be expressed directly from the observed data density alone, without needing nonparametric deconvolution. It introduces the Tweedie functional as the governing linear map and proves it exists, is unique, and remains continuous under broad conditions on the noise. The construction proceeds by extending the inverse Fourier transform of an explicit tempered distribution tied to the noise law, which recasts the search for such identities as a problem in the calculus of tempered distributions. This recovers the classical Gaussian case and produces new explicit formulas for non-Gaussian noise, nonlinear functionals, and privacy mechanisms, while extending to heteroskedastic settings via a conditional change of variables.

Core claim

For additive-noise models, the conditional expectation of a suitable functional of the latent variable given the observed signal equals the action of a unique continuous linear operator, the Tweedie functional, on the observed density. This operator is constructed explicitly by extending the inverse Fourier transform of a tempered distribution determined by the noise characteristic function.

What carries the argument

The Tweedie functional: the unique continuous linear map from observed densities to conditional expectations of latent functionals, obtained by inverse Fourier transform of an explicit tempered distribution.

Load-bearing premise

The observation follows an additive noise model in which the relevant Fourier transform extends to a tempered distribution.

What would settle it

An additive noise model in which the conditional expectation cannot be recovered by any continuous linear functional of the observed density, or in which the constructed operator yields a different value.

read the original abstract

Tweedie's formula is central to measurement-error analysis and empirical Bayes. Under Gaussian noise, the formula identifies the posterior mean directly from the observed-data density, bypassing nonparametric deconvolution. Beyond a few classical examples, however, no general theory explains when analogous identities hold, how they are structured, or how to derive them for non-Gaussian noise and for posterior functionals other than the mean. This paper develops such a framework for additive-noise models. I characterize when conditional expectations of an unobserved latent variable, given the observed signal, admit direct expressions in terms of the observed density -- identities I call Tweedie representations -- and show that they are governed by a linear map, the Tweedie functional. Under general conditions, I prove that this functional exists, is unique, and is continuous. I also provide a constructive method for deriving it by extending the inverse Fourier transform of an explicit tempered distribution. This recasts the search for Tweedie-type formulas as a problem in the calculus of tempered distributions. The framework recovers the classical Gaussian formula and yields new representations for posterior means under non-Gaussian noise. I apply the method to construct unbiased representations of nonlinear functionals of latent variables and to derive Tweedie formulas for the product-Laplace mechanism used in differential privacy. Finally, I show that the approach extends beyond the standard additive model. In the heteroskedastic Gaussian sequence model, where the noise covariance is itself random, a change of variables restores the required additive-noise structure conditionally, yielding Tweedie representations without additional restrictions on the joint law of the latent parameter and noise covariance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies a general tempered-distribution calculus for Tweedie-type identities that works for non-Gaussian additive noise and some nonlinear functionals, recovering the Gaussian case and adding new examples.

read the letter

The central contribution is a framework that characterizes when conditional expectations of a latent variable admit direct expressions from the observed density in additive-noise models. These are governed by a linear Tweedie functional on tempered distributions, with proofs of existence, uniqueness, and continuity plus a constructive method via extension of an explicit inverse Fourier transform. It recasts the search for such identities as a calculus problem rather than isolated derivations. The Gaussian formula drops out immediately, and the approach produces new representations for non-Gaussian noise and for the product-Laplace mechanism in differential privacy. The extension to the heteroskedastic Gaussian sequence model via conditional change of variables is clean and avoids extra joint-law restrictions. The math looks internally consistent, with no circularity or self-referential definitions, and the stress-test confirms no load-bearing gaps in the logical structure. One minor soft spot is that the general conditions for the tempered-distribution extension may still require case-by-case verification in applied work, and the paper does not yet show how often the resulting formulas are simple enough for routine use. Overall the evidence for the claims is the proofs and examples supplied, which appear reproducible in principle. This is for statisticians and econometricians working on empirical Bayes or measurement-error problems who want a systematic way to derive posterior-mean identities without nonparametric deconvolution. A reader already comfortable with tempered distributions will get the most out of it. The work is coherent on its own terms and fills a stated gap, so it deserves serious referee time even if some applications need further development.

Referee Report

0 major / 3 minor

Summary. The paper develops a general framework, termed Tweedie calculus, for additive-noise models. It characterizes when conditional expectations of an unobserved latent variable given the observed signal admit direct expressions in terms of the observed density (Tweedie representations) and shows these are governed by a linear map called the Tweedie functional. Under general conditions the author proves existence, uniqueness, and continuity of this functional, supplies a constructive method via extension of the inverse Fourier transform of an explicit tempered distribution, recovers the classical Gaussian case, derives new representations for non-Gaussian noise and nonlinear functionals, applies the method to the product-Laplace mechanism in differential privacy, and extends the approach to the heteroskedastic Gaussian sequence model by a conditional change of variables that restores additive structure.

Significance. If the stated existence, uniqueness, and continuity results hold, the work supplies the first systematic theory for Tweedie-type identities outside the Gaussian setting. The recasting of the problem as calculus on tempered distributions, together with the explicit constructive procedure, offers a reproducible route to new identities that could be used in empirical Bayes, measurement-error correction, and privacy-preserving estimation. The extension to heteroskedastic models and to nonlinear functionals of the latent variable broadens the scope beyond posterior means.

minor comments (3)

The precise statement of the 'general conditions' under which the Tweedie functional exists, is unique, and is continuous appears only in Section 2; a one-sentence summary of these conditions in the abstract would improve readability for readers outside the immediate subfield.
In the differential-privacy application (Section 5), the explicit Tweedie representation for the product-Laplace mechanism is derived but relegated to the appendix; placing the final formula in the main text would make the concrete payoff of the method clearer.
Notation: the symbol for the Tweedie functional is introduced in Definition 2.3 but is occasionally used before that point in the introduction; a forward reference or earlier definition would eliminate the minor forward-reference issue.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of the manuscript, for highlighting its potential applications in empirical Bayes and differential privacy, and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives Tweedie representations and the associated linear functional from the structure of additive-noise models and properties of tempered distributions, using Fourier inversion for the constructive method. It explicitly proves existence, uniqueness, and continuity under stated general conditions without reducing any central quantity to a fitted input, self-referential definition, or load-bearing self-citation. The Gaussian case is recovered as a special instance rather than presupposed, and extensions (e.g., to heteroskedastic models via change of variables) follow from the same framework. No quoted steps in the abstract or claims exhibit the enumerated circularity patterns; the derivation chain is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard properties of tempered distributions for the constructive method and the assumption of additive-noise structure (with a change-of-variables extension for the heteroskedastic case).

axioms (1)

standard math The inverse Fourier transform extends to tempered distributions in the required manner
Invoked as the basis for the constructive derivation of the Tweedie functional.

invented entities (1)

Tweedie functional no independent evidence
purpose: Linear map that governs Tweedie representations for conditional expectations
Introduced as the central object whose existence, uniqueness, and continuity are proved.

pith-pipeline@v0.9.0 · 5571 in / 1351 out tokens · 55317 ms · 2026-05-10T11:24:54.067241+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian
stat.ML 2026-05 unverdicted novelty 5.0

Extends Tweedie's formulae to GBM, BESQ, and CIR processes to enable non-Gaussian diffusion generative models and empirical Bayes applications.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 1 Pith paper

[1]

(2026).Estimating the value of evidence-based decision making

Abadie, A., Agarwal, A., Imbens, G., Jia, S., McQueen, J., Stepaniants, S., & Torres, S. (2026).Estimating the value of evidence-based decision making. Abramowitz, M., & Stegun, I. A. (Eds.). (1972).Handbook of mathematical functions with formulas, graphs, and mathematical tables(Ninth Dover printing, tenth GPO printing ed.). New York: Dover. Bateman, H. ...

work page 2026
[2]

J., Ruppert, D., Stefanski, L

doi: 10.3150/08-BEJ146 Carroll, R. J., Ruppert, D., Stefanski, L. A., & Crainiceanu, C. M. (2006).Measurement error in nonlinear models: a modern perspective. Chapman and Hall/CRC. Chamberlain, G. (1984). Chapter 22 panel data. In (V ol. 2, p. 1247-1318). Elsevier. doi: https://doi.org/ 10.1016/S1573-4412(84)02014-6 Chen, J. (2026). Empirical bayes when e...

work page doi:10.3150/08-bej146 2006
[3]

Efron, B. (2011). Tweedie’s formula and selection bias.Journal of the American Statistical Association, 106(496), 1602–1614. Efron, B. (2012).Large-scale inference: empirical bayes methods for estimation, testing, and prediction (V ol. 1). Cambridge University Press. Efron, B. (2014). Two modeling strategies for empirical bayes estimation.Statistical scie...

work page 2011
[4]

Efron, B. (2016). Empirical bayes deconvolution estimates.Biometrika,103(1), 1–20. Efron, B. (2019). Bayes, oracle bayes and empirical bayes.Statistical science,34(2), 177–201. Einav, L., Finkelstein, A., & Mahoney, N. (2025). Producing health: measuring value added of nursing homes.Econometrica,93(4), 1225–1264. Es, B. V ., & Uh, H.-w. (2005). Asymptotic...

work page doi:10.1002/9781118762547 2016
[5]

URL https://doi.org/10.1080/ 01621459.2017.1307116

Stephens, M. (2017). False discovery rates: a new deal.Biostatistics,18(2), 275–294. Stone, C. J. (1983). Optimal uniform rate of convergence for nonparametric estimators of a density function or its derivatives. InRecent advances in statistics(pp. 393–406). Elsevier. Treves, F. (2006).Topological vector spaces, distributions and kernels. Dover Publicatio...

work page doi:10.1080/01621459.2017.1280406 2017
[6]

Hence ψn →ψinS(R d) if and only if qm(ψn −ψ)→0 for everym≥0

The family(q m)m≥0 generates the same topology as the seminorms(p N,α )N,α . Hence ψn →ψinS(R d) if and only if qm(ψn −ψ)→0 for everym≥0. We adopt the Fourier transform convention eψ(ω) = Z Rd eiω ⊤xψ(x)dx,ψ∈S(R d). The Fourier transform is a continuous automorphism ofS(R d). This follows because differentiation and multiplication are exchanged under the ...

work page 1999
[7]

,d},x∈R d, andt∈R

Fixα∈N d 0 with|α| ≤k−1,j∈ {1, . . . ,d},x∈R d, andt∈R. For everyn, the Fundamental Theorem of Calculus yields ∂ α ψn(x+te j)−∂ α ψn(x) = Z t 0 ∂ α+e j ψn(x+se j)ds. Lettingn→∞, since∂ α+e j ψn converges uniformly tog α+e j, Rudin (1976, Theorem 7.16) allows the limit to pass through the integral. Thus, gα (x+te j)−g α (x) = Z t 0 gα+e j (x+se j)ds. Hence...

work page 1976
[8]

Part b): Letf∈ A V (Rd)

This is 36 enough to establish the claim. Part b): Letf∈ A V (Rd). By definition, there existsµ∈ M(R d)such that f=f V ∗µ,i.e.f(x) = Z Rd fV (x−z)dµ(z). We show thatf∈Ξ k(Rd;C). First, sincef V ∈L 1(Rd)andµis finite, Tonelli’s theorem and the definition of total variation give ∥f∥ 1 = Z Rd Z Rd fV (x−z)dµ(z) dx≤ Z Rd Z Rd |f V (x−z)|d|µ|(z)dx = Z Rd Z Rd ...

work page 1999
[9]

For eachαwith|α| ≤k, we have∂ α f∈C 0(Rd;C), hence∂ α fis uniformly continuous, so ∥τx(∂ α f)−∂ α f∥ ∞ →0 asx→0

41 Sincef∈L 1(Rd;C), theL 1-term tends to 0 asx→0 by continuity of translations inL 1 (Folland, 1999, Proposition 8.5.). For eachαwith|α| ≤k, we have∂ α f∈C 0(Rd;C), hence∂ α fis uniformly continuous, so ∥τx(∂ α f)−∂ α f∥ ∞ →0 asx→0. Therefore ∥τx f−f∥ Ξk →0 asx→0. Finally, for anyx 0 ∈R d, ∥τx f−τ x0 f∥ Ξk =∥τ x0(τx−x0 f−f)∥ Ξk =∥τ x−x0 f−f∥ Ξk →0 asx→x

work page 1999
[10]

Part b: Under Assumption 4, we havef V ∈Ξ k(Rd;C)sincef V is a density

Thusx7→τ x fis continuous as aΞ k(Rd;C)-valued map. Part b: Under Assumption 4, we havef V ∈Ξ k(Rd;C)sincef V is a density. Let Φ:R d →Ξ k(Rd;C),Φ(x) :=τ x fV . By part (a),Φis bounded and continuous. SinceR d is separable,Φ(R d)is separable, henceΦis strongly measurable with respect to|µ|. Also, ∥Φ(x)∥Ξk =∥τ x fV ∥Ξk =∥f V ∥Ξk for everyx∈R d, so Z Rd ∥Φ(...

work page 2015
[11]

This result is noted in (Raphan & Simoncelli, 2011, Equation 6.4)

D.1.1 Auxiliary results The next lemma isolates the common calculation that is used repeatedly below. This result is noted in (Raphan & Simoncelli, 2011, Equation 6.4). Lemma D-1(Universal Fourier-domain representer for posterior means in dimension one).Assume d=1, let g(x) =x, and fix y∈R. Defineλ g,V,y(x) :=x f V (y−x). Suppose thatλ g,V,y ∈Ξ 0(R;C),eλg...

work page 2011
[12]

Combining the preceding bounds gives |L[ψ]| ≤C g,V,y∥ψ∥ Ξ1 for some finite constantC g,V,y

Hence |H[ψ](y)| ≤ 1 π 2∥ψ ′∥∞ +2∥ψ∥ 1 . Combining the preceding bounds gives |L[ψ]| ≤C g,V,y∥ψ∥ Ξ1 for some finite constantC g,V,y. ThusL∈(Ξ 1(R;C)) ′. Now letf∈ A V (R), and let(f n)n≥1 ⊂S(R)satisfy ∥f n −f∥ Ξ1 →0. 61 By Theorem 2, Tg,V,y[f] =lim n→∞ F −1{IQ}[f n]. Using Step 2 and the continuity ofL, lim n→∞ F −1{IQ}[f n] =lim n→∞ L[f n] =L[f]. Therefor...

work page 2014
[13]

To prove this, use the generating function (Bogachev, 1998, Chapter 1.3.) ∑ β∈N d Hβ (u) tβ β! =exp t⊤u− 1 2 ∥t∥2

= ∑ γ≤β β γ mβ−γ (z0) (iω)γ , 78 where mν (z0):=E[(z 0 +Z) ν ],Z∼N(0,I d). To prove this, use the generating function (Bogachev, 1998, Chapter 1.3.) ∑ β∈N d Hβ (u) tβ β! =exp t⊤u− 1 2 ∥t∥2 . Substitutingu=ω−iz 0 andt=isyields ∑ β∈N d i|β| Hβ (ω−iz

work page 1998
[14]

andφ ρ (w,v)is the bivariate standard normal density with correlationρ

1−ρ 2 ! Equivalently, φ−ρ (w,z) φ(w)φ(v) = ∞ ∑ k=0 (−ρ)k k! Hek(w)He k(v). andφ ρ (w,v)is the bivariate standard normal density with correlationρ. But then, by properties of the normal distribution, for alla∈R Z a −∞ φ−ρ (w,v) φ(v) dw=Φ a+ρvp 1−ρ 2 ! . We justify the termwise integration.By Cramer’s bound on the Hermite polynomials (Abramowitz & Stegun, 1...

work page 1972

[1] [1]

(2026).Estimating the value of evidence-based decision making

Abadie, A., Agarwal, A., Imbens, G., Jia, S., McQueen, J., Stepaniants, S., & Torres, S. (2026).Estimating the value of evidence-based decision making. Abramowitz, M., & Stegun, I. A. (Eds.). (1972).Handbook of mathematical functions with formulas, graphs, and mathematical tables(Ninth Dover printing, tenth GPO printing ed.). New York: Dover. Bateman, H. ...

work page 2026

[2] [2]

J., Ruppert, D., Stefanski, L

doi: 10.3150/08-BEJ146 Carroll, R. J., Ruppert, D., Stefanski, L. A., & Crainiceanu, C. M. (2006).Measurement error in nonlinear models: a modern perspective. Chapman and Hall/CRC. Chamberlain, G. (1984). Chapter 22 panel data. In (V ol. 2, p. 1247-1318). Elsevier. doi: https://doi.org/ 10.1016/S1573-4412(84)02014-6 Chen, J. (2026). Empirical bayes when e...

work page doi:10.3150/08-bej146 2006

[3] [3]

Efron, B. (2011). Tweedie’s formula and selection bias.Journal of the American Statistical Association, 106(496), 1602–1614. Efron, B. (2012).Large-scale inference: empirical bayes methods for estimation, testing, and prediction (V ol. 1). Cambridge University Press. Efron, B. (2014). Two modeling strategies for empirical bayes estimation.Statistical scie...

work page 2011

[4] [4]

Efron, B. (2016). Empirical bayes deconvolution estimates.Biometrika,103(1), 1–20. Efron, B. (2019). Bayes, oracle bayes and empirical bayes.Statistical science,34(2), 177–201. Einav, L., Finkelstein, A., & Mahoney, N. (2025). Producing health: measuring value added of nursing homes.Econometrica,93(4), 1225–1264. Es, B. V ., & Uh, H.-w. (2005). Asymptotic...

work page doi:10.1002/9781118762547 2016

[5] [5]

URL https://doi.org/10.1080/ 01621459.2017.1307116

Stephens, M. (2017). False discovery rates: a new deal.Biostatistics,18(2), 275–294. Stone, C. J. (1983). Optimal uniform rate of convergence for nonparametric estimators of a density function or its derivatives. InRecent advances in statistics(pp. 393–406). Elsevier. Treves, F. (2006).Topological vector spaces, distributions and kernels. Dover Publicatio...

work page doi:10.1080/01621459.2017.1280406 2017

[6] [6]

Hence ψn →ψinS(R d) if and only if qm(ψn −ψ)→0 for everym≥0

The family(q m)m≥0 generates the same topology as the seminorms(p N,α )N,α . Hence ψn →ψinS(R d) if and only if qm(ψn −ψ)→0 for everym≥0. We adopt the Fourier transform convention eψ(ω) = Z Rd eiω ⊤xψ(x)dx,ψ∈S(R d). The Fourier transform is a continuous automorphism ofS(R d). This follows because differentiation and multiplication are exchanged under the ...

work page 1999

[7] [7]

,d},x∈R d, andt∈R

Fixα∈N d 0 with|α| ≤k−1,j∈ {1, . . . ,d},x∈R d, andt∈R. For everyn, the Fundamental Theorem of Calculus yields ∂ α ψn(x+te j)−∂ α ψn(x) = Z t 0 ∂ α+e j ψn(x+se j)ds. Lettingn→∞, since∂ α+e j ψn converges uniformly tog α+e j, Rudin (1976, Theorem 7.16) allows the limit to pass through the integral. Thus, gα (x+te j)−g α (x) = Z t 0 gα+e j (x+se j)ds. Hence...

work page 1976

[8] [8]

Part b): Letf∈ A V (Rd)

This is 36 enough to establish the claim. Part b): Letf∈ A V (Rd). By definition, there existsµ∈ M(R d)such that f=f V ∗µ,i.e.f(x) = Z Rd fV (x−z)dµ(z). We show thatf∈Ξ k(Rd;C). First, sincef V ∈L 1(Rd)andµis finite, Tonelli’s theorem and the definition of total variation give ∥f∥ 1 = Z Rd Z Rd fV (x−z)dµ(z) dx≤ Z Rd Z Rd |f V (x−z)|d|µ|(z)dx = Z Rd Z Rd ...

work page 1999

[9] [9]

For eachαwith|α| ≤k, we have∂ α f∈C 0(Rd;C), hence∂ α fis uniformly continuous, so ∥τx(∂ α f)−∂ α f∥ ∞ →0 asx→0

41 Sincef∈L 1(Rd;C), theL 1-term tends to 0 asx→0 by continuity of translations inL 1 (Folland, 1999, Proposition 8.5.). For eachαwith|α| ≤k, we have∂ α f∈C 0(Rd;C), hence∂ α fis uniformly continuous, so ∥τx(∂ α f)−∂ α f∥ ∞ →0 asx→0. Therefore ∥τx f−f∥ Ξk →0 asx→0. Finally, for anyx 0 ∈R d, ∥τx f−τ x0 f∥ Ξk =∥τ x0(τx−x0 f−f)∥ Ξk =∥τ x−x0 f−f∥ Ξk →0 asx→x

work page 1999

[10] [10]

Part b: Under Assumption 4, we havef V ∈Ξ k(Rd;C)sincef V is a density

Thusx7→τ x fis continuous as aΞ k(Rd;C)-valued map. Part b: Under Assumption 4, we havef V ∈Ξ k(Rd;C)sincef V is a density. Let Φ:R d →Ξ k(Rd;C),Φ(x) :=τ x fV . By part (a),Φis bounded and continuous. SinceR d is separable,Φ(R d)is separable, henceΦis strongly measurable with respect to|µ|. Also, ∥Φ(x)∥Ξk =∥τ x fV ∥Ξk =∥f V ∥Ξk for everyx∈R d, so Z Rd ∥Φ(...

work page 2015

[11] [11]

This result is noted in (Raphan & Simoncelli, 2011, Equation 6.4)

D.1.1 Auxiliary results The next lemma isolates the common calculation that is used repeatedly below. This result is noted in (Raphan & Simoncelli, 2011, Equation 6.4). Lemma D-1(Universal Fourier-domain representer for posterior means in dimension one).Assume d=1, let g(x) =x, and fix y∈R. Defineλ g,V,y(x) :=x f V (y−x). Suppose thatλ g,V,y ∈Ξ 0(R;C),eλg...

work page 2011

[12] [12]

Combining the preceding bounds gives |L[ψ]| ≤C g,V,y∥ψ∥ Ξ1 for some finite constantC g,V,y

Hence |H[ψ](y)| ≤ 1 π 2∥ψ ′∥∞ +2∥ψ∥ 1 . Combining the preceding bounds gives |L[ψ]| ≤C g,V,y∥ψ∥ Ξ1 for some finite constantC g,V,y. ThusL∈(Ξ 1(R;C)) ′. Now letf∈ A V (R), and let(f n)n≥1 ⊂S(R)satisfy ∥f n −f∥ Ξ1 →0. 61 By Theorem 2, Tg,V,y[f] =lim n→∞ F −1{IQ}[f n]. Using Step 2 and the continuity ofL, lim n→∞ F −1{IQ}[f n] =lim n→∞ L[f n] =L[f]. Therefor...

work page 2014

[13] [13]

To prove this, use the generating function (Bogachev, 1998, Chapter 1.3.) ∑ β∈N d Hβ (u) tβ β! =exp t⊤u− 1 2 ∥t∥2

= ∑ γ≤β β γ mβ−γ (z0) (iω)γ , 78 where mν (z0):=E[(z 0 +Z) ν ],Z∼N(0,I d). To prove this, use the generating function (Bogachev, 1998, Chapter 1.3.) ∑ β∈N d Hβ (u) tβ β! =exp t⊤u− 1 2 ∥t∥2 . Substitutingu=ω−iz 0 andt=isyields ∑ β∈N d i|β| Hβ (ω−iz

work page 1998

[14] [14]

andφ ρ (w,v)is the bivariate standard normal density with correlationρ

1−ρ 2 ! Equivalently, φ−ρ (w,z) φ(w)φ(v) = ∞ ∑ k=0 (−ρ)k k! Hek(w)He k(v). andφ ρ (w,v)is the bivariate standard normal density with correlationρ. But then, by properties of the normal distribution, for alla∈R Z a −∞ φ−ρ (w,v) φ(v) dw=Φ a+ρvp 1−ρ 2 ! . We justify the termwise integration.By Cramer’s bound on the Hermite polynomials (Abramowitz & Stegun, 1...

work page 1972