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11 - Convergence of Laws on Separable Metric Spaces

Published online by Cambridge University Press:  06 July 2010

R. M. Dudley
R. M. Dudley
Affiliation:
Massachusetts Institute of Technology
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Summary

So far we have dealt with convergence of laws mainly on finite-dimensional Euclidean spaces ℝk, for the central limit theorem (§§9.3–9.5). Now we'll treat converging laws on more general, possibly infinite-dimensional spaces. Here are some cases where such spaces and laws can be helpful.

Let x(t, Ω) be the position of a randomly moving particle at time t, where Ω ∈ Ω, for some probability space (Ω, , P). For each Ω, we then have a continuous function t↦ x(t, Ω). Suppose we consider times t with 0 ≤ t ≤ 1 and that x is real-valued (the particle is moving along a line, or we just consider one coordinate of its position). Then x(·, Ω) belongs to the space C[0, 1] of continuous real-valued functions on [0, 1]. The space C[0, 1] has a usual norm, the supremum norm |f| ≔, sup|f(t)|: 0 < t < 1. Then C[0, 1] is a complete separable metric space for the metric d defined as usual by d(f, g) ≔, |f – g|. It may be useful to approximate the process x, for example, by a process yn such that for each Ω and each k = 1, …, n, yn (·, Ω) is linear on the interval [(k – 1)/n, k/n]. Thus it may help to define yn converging to x in law (or in probability or a.s.) in C[0, 1].

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Real Analysis and Probability , pp. 385 - 438
Publisher: Cambridge University Press
Print publication year: 2002

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