Stein's Method and Stochastic Analysis of Rademacher Functionals
Stein's Method and Stochastic Analysis of Rademacher Functionals
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DOI：
10.1214/ejp.v15-823
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发表时间：
2008-10
影响因子：
1.4
通讯作者：
I. Nourdin;G. Peccati;G. Reinert
中科院分区：
文献类型：
--
作者：
I. Nourdin;G. Peccati;G. Reinert
我们计算显式的边界在高斯近似的泛函的无限Rademacher序列。我们的工具涉及斯坦的方法，以及使用适当的离散Malliavin运营商。由于边界是根据Malliavin算子给出的，因此不需要耦合构造。当泛函仅依赖于Rademacher序列的第一个d坐标时，得到了收敛于正态分布的一个简单的充分条件.对于有限二次型，我们得到了充分必要条件。虽然我们的方法不需要经典的使用可交换对，当功能只依赖于第一个d坐标的Rademacher序列，我们采用混沌扩展，以构建一个明确的可交换对向量的元素有关的和在混沌分解和满足线性条件的条件期望。在几个例子中，如随机变量依赖于无穷多个Rademacher变量，我们提供了三个主要的应用：（一）CLTs的多线性形式属于一个固定的混乱，（二）高斯近似加权无限2运行，（三）计算显式界限CLTs的多个积分稀疏集。这最后一个应用程序提供了一个替代的证明（和几个改进）最近的结果由布莱和詹森。
We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. As the bounds are given in terms of Malliavin operators, no coupling construction is required. When the functional depends only on the first d coordinates of the Rademacher sequence, a simple sufficient condition for convergence to a normal distribution is derived. For finite quadratic forms, we obtain necessary and sufficient conditions. Although our approach does not require the classical use of exchangeable pairs, when the functional depends only on the first d coordinates of the Rademacher sequence we employ chaos expansion in order to construct an explicit exchangeable pair vector; the elements of the vector relate to the summands in the chaos decomposition and satisfy a linearity condition for the conditional expectation. Among several examples, such as random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) to CLTs for multilinear forms belonging to a fixed chaos, (ii) to the Gaussian approximation of weighted infinite 2-runs, and (iii) to the computation of explicit bounds in CLTs for multiple integrals over sparse sets. This last application provides an alternate proof (and several refinements) of a recent result by Blei and Janson.