The modes of convergence introduced in Ch. 12 are studied in detail. Conditions for almost‐sure convergence are derived via the Borel‐Cantelli lemma. Convergence in probability is contrasted, and then a number of results for convergence of transformed series are given. Convergence in LP ‐norm is introduced as a sufficient condition for convergence in probability. Examples are given, and the chapter concludes with a preliminary look at the laws of large numbers.
Keywords: almost‐sure convergence, Borel's normal number theorem, Borel‐Cantelli lemma, convergence in LP‐norm, convergence in mean square, convergence in probability, law of large numbers, Slutsky's theorem
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