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Erschienen in:
2014 | OriginalPaper | Buchkapitel
15. Characteristic Functions and the Central Limit Theorem
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Abstract
The main goal of this chapter is the central limit theorem (CLT) for sums of independent random variables (Theorem 15.37) and for independent arrays of random variables (Lindeberg–Feller theorem, Theorem 15.43). For the latter, we prove only that one of the two implications (Lindeberg’s theorem) that is of interest in the applications.
The ideal tools for the treatment of central limit theorems are so-called characteristic functions; that is, Fourier transforms of probability measures. We start with a more general treatment of classes of test functions that are suitable to characterize weak convergence and then study Fourier transforms in greater detail. The subsequent section proves the CLT for real-valued random variables by means of characteristic functions. In the fifth section, we prove a multidimensional version of the CLT.