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On the convergence of least squares estimates in white noise

机译：关于白噪声中最小二乘估计的收敛

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摘要

The problem of convergence of least squares (LS) estimates in a stochastic linear regression model with white noise is considered. It is well known that if the parameter estimates are known to converge, the convergence analysis for many adaptive systems can be rendered considerably less arduous. For an important case where the regression vector is a measurable function of the observations and the noise is Gaussian, it has been shown, by using a Bayesian embedding argument, that the LS estimates converge almost surely for almost all true parameters in the parameter space except for a zero-measure set. However, nothing can be said about a particular given system, which is usually the objective. It has long been conjectured that such a “bad” zero measure set in the parameter space does not actually exist. A conclusive answer to this important question is provided and it is shown that the set can indeed exist. This then shows that to provide conclusive convergence results for stochastic adaptive systems, it is necessary to resort to a sample pathwise analysis instead of the Bayesian embedding approach
机译：考虑了带有白噪声的随机线性回归模型中最小二乘（LS）估计的收敛问题。众所周知，如果已知参数估计会收敛，则可以使许多自适应系统的收敛分析变得相当不费力。对于回归向量是观测值的可测量函数且噪声是高斯的重要情况，通过使用贝叶斯嵌入参数已表明，对于参数空间中几乎所有真实参数，LS估计几乎可以收敛。对于零度量集。但是，对于特定的给定系统，这通常是目标。长期以来人们一直猜测，在参数空间中设置的“坏”零度量实际上并不存在。提供了对这个重要问题的结论性答案，并表明该集合确实可以存在。然后，这表明，要为随机自适应系统提供确定的收敛结果，有必要采用样本路径分析而不是贝叶斯嵌入方法

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