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This chapter provides an overview on independence, in particular on the connections between three levels of independence: independence of a family of sets, of a family of random variables, and independence of \(\sigma \) -algebras. The existence of a sequence of independent random variables is shown by constructing the probability space carrying such a sequence. This involves the product of probability spaces and the proof of the existence of a product measure. As applications Kolmogorov’s 0-1 law and Hewitt-Savage’s 0-1 law are shown.

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Independence

  • 章节

  • 发表日期：

    2025年

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  • Hannah Geiss,
  • Stefan Geiss

摘要

This chapter provides an overview on independence, in particular on the connections between three levels of independence: independence of a family of sets, of a family of random variables, and independence of \(\sigma \) -algebras. The existence of a sequence of independent random variables is shown by constructing the probability space carrying such a sequence. This involves the product of probability spaces and the proof of the existence of a product measure. As applications Kolmogorov’s 0-1 law and Hewitt-Savage’s 0-1 law are shown.