Abstract
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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Notes
- 1.
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.
- 2.
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.
- 3.
See Parthasarathy, K.R. (1967), Theorem 6.5, pp. 46–47, or Bhattacharya and Majumdar (2007), Theorem C11.6, p. 237.
- 4.
The notations \(X_t\), \(B_t\), X(t), B(t) are all common and used freely in this text.
- 5.
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.
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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
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DOI: https://doi.org/10.1007/978-3-319-47974-3_7
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