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 asweak convergence orconvergence in distribution.

1. 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. 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. 3.

   See Parthasarathy, K.R. (1967), Theorem 6.5, pp. 46–47, or Bhattacharya and Majumdar (2007), Theorem C11.6, p. 237.

4. 4.

   The notations\(X_t\),\(B_t\),X(t),B(t) are all common and used freely in this text.

5. 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.

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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