Abstract
The normal convergency of simulated stationary random processes using the spectral representation theorem is studied. Recently, some proofs on the normality have been shown, but some incompleteness can be pointed out by considering the central limit theorem. Meanwhile, the convergency has not been studied in detail and, theoretical convergence bounds and convergence rate are not well examined. Thus, an incorrect assumption has been utilized in the previous studies. In this paper, the proofs in the previous studies are complemented with the aid of the central limit theorem, and a new proof is given based on the infinitesimal condition. Moreover, new formulae for the evaluation of convergence bound and convergence rate are derived from the Edgeworth expansion. Through the comparison with some examples, it is shown that the accuracy of new formulae is better than those in the classical central limit theorem. And also, a formula to decide the number of independent random variables is derived and it is shown that the formula satisfies any desired convergence bound. Additionally, a necessary convergence bound for the application of the Monte Carlo method based on spectral representation into extreme value problems is proposed.
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