ABSTRACT
We begin by studying the relationships among four distinct modes of convergence of a sequence of random vectors to a limit. All convergences are defined for d-dimensional random vectors. For a random vector X = (X 1,…, X d ) ∊ ℝ d the distribution function of X, defined for x = (x 1,…, x d ) ∊ ℝ d is denoted by F x(x) = P(X ≤ x) = P( X 1 ≤ x 1,…, X d ≤ x d ). The Euclidean norm of x = (x 1,…,x d ) ∊ ℝ d is denoted by |x| = https://www.w3.org/1998/Math/MathML"> ( x 1 2 + ⋯ + x d 2 ) 1 / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136288/c12c5089-77f8-4e14-84a1-364a5e26ae30/content/eq1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Let X, X 1, X 2,… be random vectors with values in ℝ d .
