Independent linear forms
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In this chapter the property of interest is the independence of linear forms in independent random variables. In Section 5.1 we give a characterization result that is both simple to state and to prove; it is nevertheless of considerable interest. Section 5.2 parallels Section 3.2. We use the characteristic property of the normal distribution to define abstract group-valued Gaussian random variables. In this broader context we again obtain the zero-one law; we also prove an important result about the existence of exponential moments. In Section 5.3 we return to characterizations, generalizing Theorem 5.1.1. We show that the stochastic independence of arbitrary two linear forms characterizes the normal distribution. We conclude the chapter with abstract Gaussian results when all forces are joined.
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