Abstract
In elementary mathematics courses (such as Calculus) one speaks of the convergence of functions: f n : R → R, then limn→∞ f n = f if limn→∞ f n (x) = f(x) for all x in R. This is called pointwise convergence of functions. A random variable is of course a function (X:Ω → R for an abstract space Ω), and thus we have the same notion: a sequence X n : Ω → R converges point-wise to X if limn→∞ X n (ω) = X(ω), for all ω ∊ Ω. This natural definition is surprisingly useless in probability. The next example gives an indication why.
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© 2000 Springer-Verlag Berlin Heidelberg
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Jacod, J., Protter, P. (2000). Convergence of Random Variables. In: Probability Essentials. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51431-9_17
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DOI: https://doi.org/10.1007/978-3-642-51431-9_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66419-2
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