On Complete Convergence And Strong Law For Weighted Sums Of I.i.d. Random Variables

In statistics, a lot of statistics are expressed in a weighted sum of random variable. In a statistical model, the key of constructing a statistics is to choose an appropriate weighted sum,such as the least squares estimate of parameters in the linear regression model,nonlinear regression model of nonparametric kernel density estimator, etc. Thus, in the context of this application, it is necessary to study the weighted sum of the sequence of random variables. Stout [1](Theorem 4.1.3, 1974) studied the limit properties of a weighted sum of random variables sequences. This type of weighted sum contains a linear regression model parameters of least squares estimate, nonlinear regression model of nonparametric kernel density estimator, etc. Therefore, the results of Stout have got a lot of attention and further research. But the subsequent results in both conditions and proof methods have no essential difference with the results of Stout. The paper improve and generalize the result of Stout(1974, Theorem 4.1.3). In particular, the sharp moment conditions are obtained and some well-known results can be obtained as special cases of the main result. And the method of the proof is completely different from in Stout’s. The main idea in the proofs of the main result is to use the invariance principle. Besides, in the strong law for weighted sums of i.i.d. random variables, the paper improves and generalizes the Li et al.(1995). Finally, the partial results are applied to the least squares.