ABSTRACT
In this chapter we introduce the definition of weak convergence of a sequence of distribution functions https://www.w3.org/1998/Math/MathML"> { F n ( x ) } n = 1 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math8_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to https://www.w3.org/1998/Math/MathML"> F ( x ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math8_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and also weak convergence of a sequence of measures https://www.w3.org/1998/Math/MathML"> { μ n } n = 1 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math8_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to https://www.w3.org/1998/Math/MathML"> μ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math8_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> For function sequences this will add to the notions of uniform and pointwise convergence of Definition I.3.40, and almost everywhere convergence in the sense of Definition I.3.17. We begin with a few examples, discussing both the distribution functions and their associated measures. Recall the discussion on distribution functions in the introduction to Chapter 6.
