Moritz Jirak
Motivated from option and derivative pricing, this note develops Edgeworth expansions both in the Kolmogorov and Wasserstein metric for many different types of discrete time volatility models and their possible transformations. This includes, among others, Hölder-type functions of (augmented) Garch processes of any order, iterated random functions or Volterra-processes.
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The validity of an approximation formula for European option prices under a general stochastic volatility model is proved in the light of the Edgeworth expansion for ergodic diffusions. The asymptotic expansion is around the Black-Scholes price and is uniform in bounded payoff func- tions. The result provides a validation of an existing singular perturbation expansion formula for the fast mean reverting stochastic volatility model.
We verify the Edgeworth expansion of any order for the integrated ergodic Levy driven Ornstein-Uhlenbeck process, applying a Malliavin calculus with truncation over the Wiener-Poisson space. Due to the special structure of the model, the coefficients of the expansion can be given in a closed form.
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points.
We study the distribution of a general class of asymptoticallylinear statistics which are symmetric functions of $N$ independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order $o(N^{-1})$. The Edgeworth expansion is based on Hoeffding's decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cramér's condition on the linear part, moment assumptions for all parts of the statistic and an…
In this work, we derive an Edgeworth expansion for the Bernoulli weighted mean $\hatμ = \frac{\sum_{i=1}^n Y_i T_i}{\sum_{i=1}^n T_i}$ in the case where $Y_1, \dots, Y_n$ are i.i.d. non semi-lattice random variables and $T_1, \dots, T_n$ are Bernoulli distributed random variables with parameter $p$. We also define the notion of a semi-lattice distribution, which gives a more geometrical equivalence to the classical Cramér's condition in dimensions bigger than 1. Our result provides a first step into the generalization of classical Edgeworth expansion theorems for random vectors that contain bo…
In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions.
We establish the validity of the empirical Edgeworth expansion (EE) for a studentized trimmed mean, under the sole condition that the underlying distribution function of the observations satisfies a local smoothness condition near the two quantiles where the trimming occurs. A simple explicit formula for the N^{-1/2} term (correcting for skewness and bias; N being the sample size) of the EE is given. In particular our result supplements previous work by P. Hall and A.R. Padmanabhan, On the bootstrap and the trimmed mean}, J. of Multivariate Analysis, v. 41 (1992), pp. 132-153. and H. Putter an…
Edgeworth expansion provides higher-order corrections to the normal approximation for a probability distribution. The classical proof of Edgeworth expansion is via characteristic functions. As a powerful method for distributional approximations, Stein's method has also been used to prove Edgeworth expansion results. However, these results assume that either the test function is smooth (which excludes indicator functions of the half line) or that the random variables are continuous (which excludes random variables having only a continuous component). Thus, how to recover the classical Edgeworth…
Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains
本文针对局部平稳马尔可夫链的部分和，证明了Berry--Esseen定理、局部中心极限定理、Edgeworth展开及大偏差原理，并给出渐近均值和方差的显式公式及最优收敛速率。
In this paper we present the Edgeworth expansion for the Euler approximation scheme of a continuous diffusion process driven by a Brownian motion. Our methodology is based upon a recent work \cite{Yoshida2013}, which establishes Edgeworth expansions associated with asymptotic mixed normality using elements of Malliavin calculus. Potential applications of our theoretical results include higher order expansions for weak and strong approximation errors associated to the Euler scheme, and for studentized version of the error process.
This paper presents new results on the Edgeworth expansion for high frequency functionals of continuous diffusion processes. We derive asymptotic expansions for weighted functionals of the Brownian motion and apply them to provide the second order Edgeworth expansion for power variation of diffusion processes. Our methodology relies on martingale embedding, Malliavin calculus and stable central limit theorems for semimartingales. Finally, we demonstrate the density expansion for studentized statistics of power variations.
While batching methods have been widely used in simulation and statistics, it is open regarding their higher-order coverage behaviors and whether one variant is better than the others in this regard. We develop techniques to obtain higher-order coverage errors for batching methods by building Edgeworth-type expansions on $t$-statistics. The coefficients in these expansions are intricate analytically, but we provide algorithms to estimate the coefficients of the $n^{-1}$ error term via Monte Carlo simulation. We provide insights on the effect of the number of batches on the coverage error where…
We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while only imposing minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail-accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized …
We prove an expansion for densities in the free CLT and apply this result to an expansion in the entropic free central limit theorem assuming a moment condition of order four for the free summands.
The one dimensional distribution of a Lévy process is not known in general even though its characteristic function is given by the famous Lévy-Khinchine theorem. This article gives an exact series representation for the one dimensional distribution of a Lévy process satisfying certain moment conditions. Moreover, this work clarifies an old result by Cramér on Edgeworth expansions for the distribution function of a Lévy process.
An operator form of asymptotic expansions for Markov chains is established. Coefficients are given explicitly. Such expansions require a certain modification of the classical spectral method. They prove to be extremely useful within the context of large deviations.
Fractional Edgeworth expansions for one-dimensional heavy-tailed random variables and applications
本文研究了一类属于 $α$-稳定随机变量吸引域的格点随机变量，它们满足截断的分数阶 Edgeworth 展开，并探讨了其运算性质、局部中心极限定理收敛速率、Green 函数展开及在离散随机偏微分方程中的应用。
An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.
We discuss sufficient conditions that guarantee the existence of asymptotic expansions for the Central Limit Theorem for weakly dependent random variables including observations arising from sufficiently chaotic dynamical systems like piece-wise expanding maps, and strongly ergodic Markov chains. We primarily use spectral techniques to obtain the results.
We derive expansions of the Hermite and Laguerre kernels at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n Laguerre Unitary Ensem- ble (LUEn), respectively. Using these large n kernel expansions, we prove an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn and LUEn. In our Edgeworth expansion, the correction terms are expressed in terms of the same Painleve II function appearing in the leading term, i.e. in the Tracy-Widom distribution. We conclude with a brief discussion of the universality of these results.
Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metr…
A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied volatilities follow. A limit theorem for at-the-money implied volatility skew and curvature is also given as a corollary. The rough Bergomi model is treated as an example.
A general partial Cramér's condition for Edgeworth expansion of a function of sample means with applications
本文提出一般偏 Cramér 条件（GPCC）以建立样本均值平滑函数分布的 Edgeworth 展开有效性，并首次验证了含连续与离散分量的 Pearson 相关系数 Edgeworth 展开的有效性，模拟表明其精度更高。
The random vector of frequencies in a generalized urn model is viewed as conditionally independent random variables, given their sum. Such a representation is exploited to derive Edgeworth expansions for a sum of functions of such frequencies. Applying these results to urn models such as with- and without-replacement sampling schemes as well as the multicolor Polya-Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without replacement scheme, and for the so-called Dixon statistic that is useful in comparing two samples.
Edgeworth expansions for centered random walks on covering graphs of polynomial volume growth
本文研究多项式体积增长覆盖图上的中心化随机游走，利用修正调和嵌入与 Trotter 逼近定理建立 Edgeworth 展开及 Berry–Esseen 型界。
In this paper we focus on the large n probability distribution function of the largest eigenvalue in the Gaussian Orthogonal Ensemble of n by n matrices (GOEn). We prove an Edgeworth type Theorem for the largest eigenvalue probability distribution function of GOEn. The correction terms to the limiting probability distribution are expressed in terms of the same Painleve II functions appearing in the Tracy-Widom distribution. We conclude with a brief discussion of the GSEn case.
Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion
本文研究独立随机变量标准化和的一阶 Edgeworth 展开逼近误差，给出任意样本量下的显式非渐近一致距离界，并推导新的 Berry–Esseen 型界及检验含义。
We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order $r$. First, the distance between the distribution of the u…
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Edgeworth type expansions of third order for transition densities are proved. This is done for time horizons that converge to 0. For this purpose we represent the transition density as a functional of densities of sums of i.i.d. variables. This will be done by application of the parametrix method. Then we apply Edgeworth expansions to the densities. The resulting series gives our Edgeworth-type expansion for the transition density of Markov chains. The research is motivated by applications to high frequ…
We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the one-split branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16(2): 886--918, 2006], Fuchs, Hwang and Neininger [Algorithmica, 46 (3--4): 367--40…
On short Edgeworth expansions for weighted sums of random vectors
本文研究 k 维独立同分布随机向量加权和的渐近行为，证明在五阶绝对矩有限时，Edgeworth 修正的收敛阶为 O(1/n^{3/2})，推广了 Bobkov 的一维结果。
Edgeworth corrections for the spiked eigenvalues of non-Gaussian sample covariance matrices with applications
本文为非高斯数据下的单尖刺和多尖刺场景建立了尖刺特征值的一阶 Edgeworth 展开，借此构建更准确的置信区间并估计尖刺数量，在稳健性和准确性上优于现有方法。
Generalized Edgeworth expansions for integer-valued additive functionals of uniformly elliptic Markov chains
本文为一致椭圆非齐次马尔可夫链的整数值加性泛函的部分和概率建立了广义Edgeworth展开，并利用三角多项式的显式公式，将标准Edgeworth展开的存在性与模等分布速率联系起来。
We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of arXiv:0803.0458 to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctu…
Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed random elements with law $μ$ on the general linear group $\textup{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$. Under suitable conditions on $μ$, we establish the first-order Edgeworth expansion for the coefficients $\langle f, G_n v \rangle$ with $v \in V$ and $f \in V^*$, in which a new additional term appears compared to the case of vector norm $\|G_n v\|$.
Local limit theorem and Edgeworth expansions for inhomogeneous random walks on $GL(d,\mathbb R)$
本文针对可逆独立随机矩阵乘积的范数对数，证明了非格型局部中心极限定理与埃奇沃思展开，还为文献[MatBE]的中心极限定理最优速率提供了新证明，并给出收缩的若干充分条件。
We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter $\varepsilon$. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in $\varepsilon$ and provide explicit expressions …
Optimal Non-Asymptotic Edgeworth Expansions for Multivariate Neural Network Outputs
针对有限宽度全连接神经网络输出，利用任意阶 Edgeworth 展开逼近其与高斯极限的偏差，并给出总变差距离的上下界。
We prove that certain asymptotic moments exist for some random distance expanding dynamical systems and Markov chains in random dynamical environment, and compute them in terms of the derivatives at the $0$ of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments $\gam_k=\lim_{n\to\infty}n^{-[\frac k2]}\bbE(\sum_{i=1}^n X_i)^k$ of sums of independent and identically distributed random variables satisfy. We will also obtain certain (Edgeworth) asymptotic expansions related to the central limit theorem for such processes. Our proofs…
Non-uniform Edgeworth expansions for weakly dependent random variables and their applications
本文首次在弱相依非平稳序列（如非齐次马尔可夫链、时变动力系统、随机矩阵乘积等）上建立了非均匀Edgeworth展开，并将其应用于分布函数的$L^p$平均估计、多项式增长函数的期望与矩展开以及Wasserstein距离下的Berry-Esseen定理。
Edgeworth expansions of extreme eigenvalue distributions for random unitary ensembles
本文建立随机酉系综极端特征值分布的Edgeworth展开，揭示修正项的普适结构，并给出最大和最小特征值分布的完全展开。
Consider a homogeneous Poisson process in $\mathbb{R}^d$, $d \ge 1$. Let $R_1 < R_2 < \dots$ be the distances of the points from the origin, and let $S = R_1^{-γ} + R_2^{-γ} + \dots$, where $γ> d$ is a parameter. Let $\overline{S}^{(r)} = \sum_k R_k^{-γ} \, \mathbf{1}_{R_k \ge r}$ be the contribution to $S$ outside radius $r$. For large enough $r$, and any $\overline{s}$ in the support of $\overline{S}^{(r)}$, consider the change of measure that shifts the mean to $\overline{s}$. We derive rigorous error estimates for the Edgeworth expansion of the transformed random variable. Our error terms …
Consider the sum $Z = \sum_{n=1}^\infty λ_n (η_n - \mathbb{E}η_n)$, where $η_n$ are i.i.d.~gamma random variables with shape parameter $r > 0$, and the $λ_n$'s are predetermined weights. We study the asymptotic behavior of the tail $\sum_{n=M}^\infty λ_n (η_n - \mathbb{E}η_n)$ which is asymptotically normal under certain conditions. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. We illustrate the effectiveness of these expansions on an infinite sum of weighted chi-squared distributions.
We prove a Berry-Esseen theorem and Edgeworth expansions for partial sums of the form $S_N=\sum_{n=1}^{N}f_n(X_n,X_{n+1})$, where $\{X_n\}$ is a uniformly elliptic inhomogeneous Markov chain and $\{f_n\}$ is a sequence of uniformly bounded functions. The Berry-Esseen theorem holds without additional assumptions, while expansions of order $1$ hold when $\{f_n\}$ is irreducible, which is an optimal condition. For higher order expansions, we then focus on two situations. The first is when the essential supremum of $f_n$ is of order $O(n^{-\be})$ for some $\be\in(0,1/2)$. In this case it turns out…
We consider a scheme of equiprobable allocation of particles into cells by sets. The Edgeworth type asymptotic expansion in the local central limit theorem for a number of empty cells left after allocation of all sets of particles is derived.
We provide asymptotic expansions for the Stirling numbers of the first kind and, more generally, the Ewens (or Karamata-Stirling) distribution. Based on these expansions, we obtain some new results on the asymptotic properties of the mode and the maximum of the Stirling numbers and the Ewens distribution. For arbitrary $θ>0$ and for all sufficiently large $n\in\mathbb N$, the unique maximum of the Ewens probability mass function $$ \mathbb L_n(k) = \frac{θ^k}{θ(θ+1)\ldots(θ+n-1)} \genfrac{[}{]}{0pt}{}{n}{k}, \quad k=1,\ldots,n, $$ is attained at $k= \left\lfloor θ\log n + \frac{θΓ'(θ)}{Γ(θ)} -…
Edgeworth expansion and large deviations for the coefficients of products of positive random matrices
本文研究正随机矩阵乘积元素的精细极限行为，在最优矩条件下建立 Berry–Esseen 定理、Edgeworth 展开及大偏差渐近，并导出局部极限定理与谱半径偏差界。
Edgeworth expansion on Wiener chaos
本文在Wiener混沌上建立了任意阶的Edgeworth展开，以carré-du-champ算子的方差为小参数，给出显式余项界，并恢复最优四阶矩定理。
An error term in the Central Limit Theorem for sums of discrete random variables
研究具有$d+1$个原子的独立同分布随机变量之和，证明其$d-1$阶Edgeworth展开几乎处处有效，且误差典型为$n^{-d/2}$阶。
Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random Fields
本文将Wasserstein-p界扩展至α-混合随机场，通过累积量Edgeworth展开与Stein方法，证明其重标平均值以O(|T|^(-β))收敛于标准正态分布，β可达1/2。
Breakdown of Edgeworth Expansion in Finite-Blocklength Regime and Exact Absorption via $q$-Deformation
本文针对有限块长 regime 下埃奇沃思展开的负概率问题，提出 q-变形框架，通过动态缩放 q-对数参数实现三阶偏度的精确吸收，为超可靠通信提供稳健基础。
Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions
本文研究高维随机环面几何图与ER图的区分问题，提出基于修正Edgeworth展开的新技术来量化边依赖性，首次给出一般L_q范数下的紧致算法上界与改进的信息论下界，并精确刻画了其谱性质。
Martingale expansion for stochastic volatility
本文提出了一种适用于连续随机波动模型的马丁格尔展开方法，能够处理小波动率-波动率和快速均值回归模型，并在最小条件下得出一阶扰动展开。
We introduce an asymptotic small noise expansion, a so called vol-of-vol expansion, for potentially infinite dimensional and rough stochastic volatility models. Thereby we extend the scope of existing results for finite dimensional models and validate claims for infinite dimensional models. Furthermore we provide new, explicit (in the sense of non-recursive) representations of the so-called push-down Malliavin weights that utilizes a precise understanding of the terms of this expansion.
Consider a branching random walk on $\mathbb Z$ in discrete time. Denote by $L_n(k)$ the number of particles at site $k\in\mathbb Z$ at time $n\in\mathbb N_0$. By the profile of the branching random walk (at time $n$) we mean the function $k\mapsto L_n(k)$. We establish the following asymptotic expansion of $L_n(k)$, as $n\to\infty$: $$ e^{-φ(0)n} L_n(k) = \frac{e^{-\frac 12 x_n^2(k)}}{\sqrt {2πφ''(0) n}} \sum_{j=0}^r \frac{F_j(x_n(k))}{n^{j/2}} + o\left(n^{-\frac{r+1}{2}}\right) \quad a.s., $$ where $r\in\mathbb N_0$ is arbitrary, $φ(β)=\log \sum_{k\in\mathbb Z} e^{βk} \mathbb E L_1(k)$ is th…
Edgeworth-type expansion for the one-point distribution of the KPZ fixed point with a large height at a prior location
本文研究KPZ不动点在先前位置大高度条件下的单点分布，证明其收敛于GUE Tracy-Widom分布并给出Edgeworth型低阶展开，与后期大高度情形的高斯型极限共同构成完整相图。
An Edgeworth-type expansion is established for the relative Fisher information distance to the class of normal distributions of sums of i.i.d. random variables, satisfying moment conditions. The validity of the central limit theorem is studied via properties of the Fisher information along convolutions.
We study a Edgeworth-type refinement of the central limit theorem for the discretizacion error of Itô integrals. Towards this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula.
The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived in Yoshida (1997) as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a m…
We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of densities to derive expansions for the free entropic distance of sums to the Wigner law.
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