Wiener空间中的双边伽马逼近
Bilateral Gamma Approximation in Weiner Space
AI总结：
本文利用基于积分算子的Malliavin-Stein方法，研究等正态高斯过程泛函到双边伽马分布的逼近，给出光滑Wasserstein距离的误差界，并应用于第二Wiener混沌、齐次和及U统计量的收敛性分析。
摘要：
本文研究等正态高斯过程泛函的双边伽马（BG）逼近。我们使用Malliavin-Stein方法获得光滑Wasserstein距离的误差界。作为副产品，给出了方差伽马（VG）、拉普拉斯、伽马和正态逼近的误差界。我们的方法的新颖之处在于Stein方程基于积分算子，而非文献中常用的微分算子。我们的一些界比现有界更锐利。对于第二Wiener混沌中随机元到BG分布的逼近，其界以累积量的形式给出。利用这一结果，我们证明第二Wiener混沌中的随机变量（rvs）序列依分布收敛于一个BG随机变量，当且仅当它们的二阶至六阶累积量收敛。作为我们结果的一个应用，我们考虑独立随机变量的齐次和到BG分布的逼近，并提及一些相关的极限定理。最后，讨论了U统计量到BG分布的逼近。
英文摘要：
This paper deals with bilateral-gamma (BG) approximation to functionals of an isonormal Gaussian process. We use Malliavin-Stein method to obtain the error bounds for the smooth Wasserstein distance. As by-products, the error bounds for variance-gamma (V G), Laplace, gamma and normal approximations are presented. Our approach is new in the sense that the Stein equation is based on integral operators rather than diferential operators commonly used in the literature. Some of our bounds are sharper than the existing ones. For the approximation of a random element from the second Wiener chaos to a BG distribution, the bounds are obtained in terms of their cumulants. Using this result, we show that a sequence of random variables (rvs) in the second Wiener chaos converges in distribution to a BG rv if their cumulants of order two to six converge. As an application of our results, we consider an approximation of homogeneous sums of independent rvs to a BG distribution, and mention some related limit theorems also. Finally, an approximation of a U-statistic to the BG distribution is discussed.
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