Part of the book series: Universitext ((UTX))
Let \((S,\rho )\) be a metric space and let \(\mathcal {P}(S)\) be the set of all probability measures on \((S, \mathcal {B}(S)).\) In this chapter we consider a general formulation of convergence in \(\mathcal {P}(S)\), referred to as weak convergence or convergence in distribution.
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Billingsley (1968) provides a detailed exposition and comprehensive account of the weak convergence theory. From the point of view of functional analysis, weak convergence is actually convergence in the weak* topology. However the abuse of terminology has become the convention in this context.
A non-Fourier analytic proof was found by Guenther Walther (1997): On a conjecture concerning a theorem of Cramér and Wold, J. Multivariate Anal., 63, 313–319, resolving some serious doubts about whether it would be possible.
See Parthasarathy, K.R. (1967), Theorem 6.5, pp. 46–47, or Bhattacharya and Majumdar (2007), Theorem C11.6, p. 237.
The notations \(X_t\), \(B_t\), X(t), B(t) are all common and used freely in this text.
While the result here is merely that convergence in the bounded-Lipschitz metric implies weak convergence, the converse is also true. For a proof of this more general result see Bhattacharya and Majumdar (2007), pp. 232–234. This metric was originally studied by Dudley, R.M. (1968): Distances of probability measures and random variables, Ann. Math. 39, 15563–1572.
© 2016 Springer International Publishing AG
Bhattacharya, R., Waymire, E.C. (2016). Weak Convergence of Probability Measures on Metric Spaces. In: A Basic Course in Probability Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-47974-3_7
DOI: https://doi.org/10.1007/978-3-319-47974-3_7
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-47972-9
Online ISBN: 978-3-319-47974-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
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