具有估计误差的多元Laplace逼近及其在极限定理中的应用
Multivariate Laplace's approximation with estimated error and application to limit theorems
摘要：
本文对指数最大值在区域内部和边界上的两种情况，给出了含大参数的多元Laplace积分的一个近似，并估计了误差项。我们特别感兴趣的情况下，在指数函数依赖于大参数。作为应用，证明了弱大数定律和中心极限定理。第二个结果给出了上述两种情况下不同的极限分布。当指数的最大值在区域内部时，它是正态分布，如果它在边界上，它在一个积分方向上是指数分布，在其他方向上是正态分布。
英文摘要：
In this paper we obtain an approximation for the multivariate Laplace's integral with a large parameter and estimate error term for two cases, when the maximum of the exponent is in the interior of the domain and on the boundary. We are specifically interested in the situation when the function in the exponent depends on the large parameter. As an application we prove weak law of large numbers and central limit theorem. The second result gives different limiting distributions for two cases mentioned above. When the maximum of the exponent is in the interior of the domain it is Normal distribution and if it is on the boundary, it is Exponential in one direction of integration and Normal in other directions.
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