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Update reference to definition of Wiener process to 5th ed. of Durrett |
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Another characterisation of a Wiener process is the [[definite integral]] (from time zero to time ''t'') of a zero mean, unit variance, delta correlated ("white") [[Gaussian process]].{{citation needed|date=October 2015}}
The Wiener process can be constructed as the [[scaling limit]] of a [[random walk]], or other discrete-time stochastic processes with stationary independent increments. This is known as [[Donsker's theorem]]. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed [[neighborhood (mathematics)|neighborhood]] of the origin infinitely often) whereas it is not recurrent in dimensions three and higher.<ref>{{
:<math>\alpha^{-1}W_{\alpha^2 t}</math>
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