Abstract
The classical central limit theorem was generalized to a functional central limit theorem by Donsker (1951) (see Theorem A.3.2). In words the result means that one considers the partial sums {S o, S 1,..., S n } of i.i.d. variables jointly for each n and shows that if the mean and variance are finite then the (polygonal) process obtained by normalization (and linear interpolation), behaves, asymptotically, like Brownian motion.
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Gut, A. (2009). Functional Limit Theorems. In: Stopped Random Walks. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87835-5_5
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DOI: https://doi.org/10.1007/978-0-387-87835-5_5
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