The asymptotic Berry-Esseen constant for intervals

Berry-Esseen Type Bound for Fractional Ornstein-Uhlenbeck Type Process Driven by Sub-fractional Brownian Motion

We obtain a Berry-Esseen type bound for the distribution of the maximum likelihood estimator of the drift parameter for fractional Ornstein-uhlenbeck type process driven by sub-fractional Brownian motion.

Berry-Esseen bounds in the entropic central limit theorem

Berry-Esseen-type bounds for total variation and relative entropy distances to the normal law are established for the sums of non-i.i.d. random variables.

网络中模块度的 Berry-Esseen 界

Berry-Esseen bound of modularity in network

针对给定网络的特定划分，研究证明了网络规模增大时模块度的 Berry-Esseen 界与强大数定律。

Berry-Esseen bound for the Parameter Estimation of Fractional Ornstein-Uhlenbeck Processes

For an Ornstein-Uhlenbeck process driven by fractional Brownian motion with Hurst index $H\in [\frac12,\frac34]$, we show the Berry-Esséen bound of the least squares estimator of the drift parameter. We use an approach based on Malliavin calculus given by Kim and Park \cite{kim 3}.

基于Malliavin-Stein方法的非均匀Berry-Esseen界

Non-uniform Berry-Esseen bounds via Malliavin-Stein method

本文利用Malliavin-Stein方法建立非均匀Berry-Esseen界，并应用于多重Wiener-Itô积分与布朗运动指数泛函。

Berry-Esseen Bounds for typical weighted sums

Under correlation-type conditions, we derive upper bounds of order $\frac{1}{\sqrt{n}}$ for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.

On a Berry-Esseen type limit theorem for Boolean convolution

We obtain a sharp estimate of the speed of convergence in the Boolean central limit theorem for measures of finite sixth moment. The main tool is a quantitative version of the Stieltjes-Perron inversion formula.

Optimal Berry-Esseen bounds on the Poisson space

We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati (2012, 2015) and Biermé, Bonami, Nourdin and Peccati (2013), involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs.

Nonuniform Berry-Esseen bounds for martingales with applications to statistical estimation

We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate $x$'s, and are of exponential decay rate as de la Peña's inequality when $x\rightarrow \infty$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.

Berry-Esseen bounds in the Breuer-Major CLT and Gebelein's inequality

We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function $φ$ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein's inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients.

A Berry Esseen type limit theorem for Boolean convolution

We give estimates on the rate of convergence in the Boolean central limit theorem for the Lévy distance. In the case of measures with bounded support we obtain a sharp estimate by giving a qualitative description of this convergence.

对可数字母表上熵和多样性指数估计量的Berry-Esseen界

Berry-Esseen bounds for estimators of entropy and diversity indices on countable alphabets

本文推导了可数字母表上多样性指数估计量的Berry-Esseen界，建立了广泛指数类插件估计量的非渐近收敛速率，包括Simpson指数和Renyi熵。针对Shannon熵的实用重要情况，给出了标准插件估计量以及两种常用偏置修正变体的显式Berry-Esseen界。

On rates of convergence and Berry-Esseen bounds for random sums of centered random variables with finite third moments

We show, how the classical Berry-Esseen theorem for normal approximation may be used to derive rates of convergence for random sums of centerd, real-valued random variables with respect to a certain class of probability metrics, including the Kolmogorov and the Wasserstein distances. This technique is applied to several examples, including the approximation by a Laplace distribution of a geometric sum of centered random variables with finite third moment, where a concrete Berry-Esseen bound is derived. This bound reduces to a bound of the supposedly optimal order $\sqrt{p}$ in the i.i.d. case.

伯德-埃森定理的一个友好的证明

Two friendly proofs of the Berry-Esseen theorem

本文通过直观的傅里叶分析方法，给出了伯德-埃森定理的非渐近形式证明，适用于基础概率课程。

条件为保持正的随机游走的Berry-Esseen不等式

Berry-Esseen inequality for random walks conditioned to stay positive

本文针对条件为保持正的随机游走，推导Berry-Esseen型估计，证明了在增量零均值、有限方差下，其收敛于瑞利分布的速率为$n^{-1/2}$阶。

Berry-Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GLd(R)

We give rates of convergence in the Central Limit Theorem for the coefficients and the spectral radius of the left random walk on GLd(R), assuming the existence of an exponential or polynomial moment.

Berry-Esseen bounds for weighted averages of Poisson avoidance functionals

We consider functionals which are weighted averages of the avoidance function of a Poisson process. Using the approach to Stein's method based on Malliavin calculus for Poisson functionals we provide explicit bounds for the Wasserstein distance between these standardized functionals and the standard normal distribution. Our approach relies on closed-form expressions for the action of some Malliavin type operators on avoidance functionals of Poisson processes. As a result we provide Berry-Esseen bounds in the CLT for the volume of the union of balls of a fixed radius around random Poisson cente…

Berry-Esseen's bound and Cramér's large deviation expansion for a supercritical branching process in a random environment

Let $(Z_n)$ be a supercritical branching process in a random environment $ξ= (ξ_n)$. We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for $\log Z_n$ under the annealed law $\mathbb P$. We also improve some earlier results about the harmonic moments of the limit variable $W=lim_{n\to \infty} W_n$, where $W_n =Z_n/ \mathbb{E}_ξ Z_n$ is the normalized population size.

de Jong中心极限定理中的Berry-Esseen界

The Berry-Esseen bound in de Jong's CLT

本文针对归一化完全退化U-统计量，在四阶矩收敛于3和Lindeberg-Feller型条件下证明了de Jong中心极限定理的Berry-Esseen界，并达到与已有Wasserstein距离界相同的最优阶。

Berry--Esseen 界对所有足够大的样本量都是精确的

The Berry-Esseen Constant Conjecture is Eventually True

本文证明在样本量足够大时，Berry--Esseen 不等式中的 Esseen 常数是最优的，并给出通用阈值 $N=2\lceil\exp(10^{17})\rceil$。

Berry-Esseen-type estimates for random variables with a sparse dependency graph

We obtain Berry-Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order $δ\in (2,\infty]$ using a Fourier transform approach. Our bounds improve the state-of-the-art in the regime where the degree of the dependency graph is large. As a Corollary of our results, we obtain a Central Limit Theorem for random variables with a sparse dependency graph that are uniformly bounded in $L^δ$ for some $δ\in(2,\infty]$.

Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences

Berry-Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin-Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small-ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.

A conditional Berry-Esseen bound and a conditional large deviation result without Laplace transform. Application to hashing with linear probing

\noindent We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. We prove a Berry-Esseen bound in a general setting and a large deviation result when the Laplace transform of the underlying distribution is not defined in a neighborhood of zero. Then we present several combinatorial applications. In particular, we prove a large deviation result for the model of hashing with linear probing.

A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph

Applying Stein's method, an inductive technique and size bias coupling yields a Berry-Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdos-Renyi random graph of a given degree.

Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes

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…

可交换对的非一致 Berry–Esseen 界及其在平均场经典 $N$-向量模型和 Jack 测度中的应用

Non-uniform Berry--Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures

本文借助 Stein 方法与集中不等式建立可交换对的非一致 Berry–Esseen 界，改进已有定理，并将其应用于平均场经典 $N$-向量模型和 Jack 测度。

Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices and consider the matrix product $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and moderate deviation expansions of Cramér type, for the matrix norm $\| G_n \|$ of $G_n$, for its $(i,j)$-th entry $G_n^{i,j}$, and the and for its spectral radius $ρ(G_n)$.

A Berry-Esseen bound with (almost) sharp dependence conditions

Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of $n^{-1/2}$ in the uniform metric, in other words, when do we have the optimal Berry-Esseen bound? We study this question in a quite general framework and find the (almost) sharp dependence conditions. The result applies to many different processes and dynamical systems. As specific, prominent examples, we study functions of the doubling map 2x mod 1 and the left random walk on the general linear group.

Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field

We study the inhomogeneous Curie-Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.

双曲空间上布朗运动的Berry-Esseen界

Berry-Esseen bound for the Brownian motions on hyperbolic spaces

本文研究了双曲空间上布朗运动径向部分的高斯波动，基于Millson公式和分部积分公式得到了均匀收敛速率，并证明在维数为2或一般奇数时该结果是精确的。

Variations on the Berry-Esseen theorem

We analyze the quality of the gaussian approximation to linear combinations of n independent, identically-distributed random variables with finite fourth moments. It turns out that there exist universal, simple linear combinations that perform better than the sum of the variables. We also investigate the case in which the random variables are independent, yet they are not necessarily identically distributed.

More on the nonuniform Berry--Esseen bound

Certain smoothing inequalities were proposed in the recent paper posted on arXiv at arxiv:1301.2828 in order to lessen the very large gap between the best correctly established upper and lower bounds on the constant factor in the nonuniform Berry-Esseen bound. As an illustration of the possible uses of those inequalities, a quick proof of Nagaev's classical nonuniform bound was given there. Here we describe another, apparently more effective class of smoothing inequalities, and give a yet quicker and simpler proof of Nagaev's result.

组合中心极限定理和模式出现的Berry Esseen界，使用零和大小偏置

Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

Berry Esseen型边界的正常，基于零和大小偏置耦合，使用斯坦的方法。零偏界的组合中心极限定理的随机置换具有均匀分布或一个是恒定的置换具有相同的周期类型，并没有不动点的应用程序进行说明。大小偏置界限被应用于固定的相对有序的子序列（如上升序列）在随机排列中的出现，以及有限图上的模式、极值和子图的出现。

On a bound of the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables

It is shown that the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\le n\le 500000$, where $n$ is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009--2011, an upper bound is obtained for the absolute constant in the Berry--Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.0…

A Berry-Esseen bound of order $ 1/\sqrt{n} $ for martingales

Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$ in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen (Ann. Probab. 1982) and Grama and Haeusler (Stochastic Process. Appl. 2000), is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.

Berry-Esseen estimates for regenerative processes under weak moment assumptions

We prove Berry-Esseen type rates of convergence for central limit theorems (CLTs) of regenerative processes which generalize previous results of Bolthausen under weaker moment assumptions. We then show how this general result can be applied to obtain rates of convergence for (1) CLTs for additive functionals of positive recurrent Markov chains under certain conditions on the strong mixing coefficients, and (2) annealed CLTs for certain ballistic random walks in random environments.

Berry-Esseen bounds for functionals of independent random variables

We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on chaos expansions methods. Our results apply to $U$-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate $U$-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weigh…

A multivariate Berry--Esseen theorem with explicit constants

We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein's method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.

Berry-Esseen type bounds for the Left Random Walk on GL d (R) under polynomial moment conditions

Let $A_n= \varepsilon_n \cdots \varepsilon_1$, where $(\varepsilon_n)_{n \geq 1}$ is a sequence of independent random matrices taking values in $ GL_d(\mathbb R)$, $d \geq 2$, with common distribution $μ$. In this paper, under standard assumptions on $μ$ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for $\log ( \Vert A_n \Vert)$ when $μ$ has a polynomial moment. More precisely, we get the rate $((\log n) / n)^{q/2-1}$ when $μ$ has a moment of order $q \in ]2,3]$ and the rate $1/ \sqrt{n} $ when $μ$ has a moment of order $4$, which significantly improves earlier r…

Central limit theorem and Berry-Esseen bounds for a branching random walk with immigration in a random environment

We consider a branching random walk on $d$-dimensional real space with immigration in a time-dependent random environment. Let $Z_n(\mathbf t)$ be the so-called partition function of the process, namely, the moment generating function of the counting measure describing the dispersion of individuals at time $n$. For $\mathbf t$ fixed, the logarithm $\log Z_n(\mathbf t)$ satisfies a central limit theorem. By studying the logarithmic moments of the intrinsic submartingale of the system and its convergence rates, we establish the uniform and non-uniform Berry-Esseen bounds corresponding to the cen…

An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$ρ(F_n,Φ)\le\frac{0.335789(β^3+0.425)}{\sqrt{n}}$$ and $$ρ(F_n,Φ)\le \frac{0.3051(β^3+1)}{\sqrt{n}} $$ are proved for the uniform distance $ρ(F_n,Φ)$ between the standard normal distribution function $Φ$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $β^3$. The first of these inequalities sharpens the best known version of the clas…

New Berry-Esseen bounds for non-linear functionals of Poisson random measures

This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations on the Poisson space, we derive a bound, which is strictly smaller than what is available in the literature. This is applied to sequences of multiple integrals and sequences of Poisson functionals having a finite chaotic expansion. This leads to new Berry-Esseen bounds in de Jong's theorem for degenerate U-statistics. Moreover, geometric functionals of inters…

Feynman-Kac模型和相互作用粒子模型的Berry-Esseen定理

A Berry-Esseen theorem for Feynman-Kac and interacting particle models

本文研究了一类Feynman-Kac粒子近似模型涨落的收敛速度。我们设计了一个新的Berry-Esseen型估计的基础上的抽象鞅序列结合原来的指数浓度估计的相互作用的过程。这些结果扩展了经典理论中的相应陈述，并适用于一类在非线性滤波文献以及统计物理和生物学中出现的分支和谱系路径粒子模型。

Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon

Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\geq 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise variance expansion for the vertex number is determined up to the constant-order term, a result which can be considered as a second-order analogue of the classical expansion for the expectation of Rényi and Sulanke (1963). Moreover, a sharp Berry-Esseen bound is derived for the vertex number of the random polygon $P_n$, which is of the same order as the square-…

AR(1) processes driven by second-chaos white noise: Berry-Esséen bounds for quadratic variation and parameter estimation

In this paper, we study the asymptotic behavior of the quadratic variation for the class of AR(1) processes driven by white noise in the second Wiener chaos. Using tools from the analysis on Wiener space, we give an upper bound for the total-variation speed of convergence to the normal law, which we apply to study the estimation of the model's mean-reversion. Simulations are performed to illustrate the theoretical results.

Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under the annealed law, uniformly in $n_0 \in \mathbb{N}$, which extend the corresponding results by Grama et al. (Stochastic Process.\ Appl. 2017) established for $n_0=0$. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. (2017) are relaxed in our present setting. An example of appl…

Berry-Esseen and Edgeworth approximations for the tail of an infinite sum of weighted gamma random variables

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.

Berry-Esseen bounds and almost sure CLT for the quadratic variation of a general Gaussian process

In this paper, we consider the explicit bound for the second-order approximation of the quadratic variation of a general fractional Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by $(ts)^{H-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or have stationary increments. %Some examples include the su…

On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands

By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities $Δ_n\leq0.3328(β_3+0.429)/\sqrt{n}$ and $Δ_n\leq0.33554(β_3+0.415)/\sqrt{n}$ are proved for the uniform distance $Δ_n$ between the standard normal distribution function and the distribution function of the normalized sum of an arbitrary number $n\geq1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $β_3$. The first of these two inequalities improves one that was proved in (Korolev and Shevtsova, 2010), and as well sharp…

关于Character比率的Berry-Esseen定理的归纳证明

An Inductive Proof of the Berry-Esseen Theorem for Character Ratios

Bolthausen使用了Stein方法的一种变体，对独立同分布随机变量的Berry-Esseen定理进行了归纳证明。我们修改这种技术证明了一个贝里-埃辛定理的随机表示的对称群的转置字符比。对于划分上的Jack测度也证明了类似的结果。

基于Malliavin-Stein方法的一维扩散过程长时间渐近性的Berry-Esseen界

Berry-Esseen bounds for large-time asymptotics of one-dimensional diffusion processes via Malliavin-Stein method

本文研究一维随机微分方程解在时间趋于无穷时的渐近行为，利用Malliavin-Stein方法证明了中心化缩放解分布与标准正态分布间全变差距离的Berry-Esseen界，并获得了时间参数的最优收敛速率。

Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs

An exchangeable pair approach is commonly taken in the normal and non-normal approximation using Stein's method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein's method, a new Berry-Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for no…

A Berry-Esseen Bound for Vector-valued Martingales

This note provides a conditional Berry-Esseen bound for the sum of a martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to a filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate the conditional distribution of $S=\sum_{i=1}^n X_i$ given some $σ$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that of a mean-zero normal random vector having the same conditional variance given $\mathcal{F}_0 $ as the vector $S$. Assuming that the conditional variances $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge 1$, are $\mathcal{F}_0$-measurable and non-singular, an…

弱相关下样本分位数的一个Berry-Esseen定理

A Berry--Esseen theorem for sample quantiles under weak dependence

本文证明了多项式混合率下强混合随机变量的样本分位数的一个Berry-Esseen定理。正态近似率显示为$O(n^{-1/2})$作为$n\to\infty$，其中$n$表示样本量。这一结果与强混合随机变量样本均值的情形形成鲜明对比，强混合随机变量样本均值的速率$O(n^{-1/2})$即使在指数强混合速率下也是未知的。由于金融时间序列数据往往是重尾的，基于分位数的方法在金融中的各种问题，包括套期保值和风险管理中发挥着重要作用，本文的主要结果在金融和计量经济学中有应用。