ABSTRACT
In this chapter we begin the development of an investigation into the probability of “tail” events. We assume that there is a probability space https://www.w3.org/1998/Math/MathML"> ( S , E , μ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math9_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in the background, and a random variable X or sequence https://www.w3.org/1998/Math/MathML"> { X 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/math9_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with associated distribution functions F and https://www.w3.org/1998/Math/MathML"> { F 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/math9_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> By a tail event of a distribution function F is meant:
Left Tail Event: Given x, the event is defined as https://www.w3.org/1998/Math/MathML"> { X ≤ x } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math9_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with associated probability https://www.w3.org/1998/Math/MathML"> Pr { X ≤ x } = μ X − 1 ( − ∞ , x ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math9_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> : https://www.w3.org/1998/Math/MathML"> Pr { X ≤ x } = 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/math9_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Right Tail Event: Given x, the event is defined as https://www.w3.org/1998/Math/MathML"> { X ≥ x } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math9_7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with associated probability https://www.w3.org/1998/Math/MathML"> Pr { X ≥ x } ≡ μ X − 1 [ x , ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math9_8.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> : https://www.w3.org/1998/Math/MathML"> Pr { X ≥ x } = 1 − 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/math9_9.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
