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Donsker's invariance principle is the primary example of a functional central limit theorem. It provides conditions under which a stochastic process converges weakly to a Brownian motion, B_t. For example, if one considers the simple random walk on the integers, S_t, after appropriate rescaling one finds that the probability distribution approaches that of a one-dimensional Brownian motion: $Ef(S_{kt}/sqrt(t)) \to Ef(B_k)$ as t goes to infinity, for appropriate functions f, k>0.
We will discuss this result before giving a (shortened) proof of a general result, characterizing similar invariance principles for diffusions other than Brownian motion. As an example, we will apply the theory to show that the weak limit of an urn model is an Ornstein-Uhlenbeck process.
A group with a cyclically symmetric presentation admits an automorphism of finite order called the shift. In this talk we look at families of nonaspherical, cyclically presented groups and we show that these give rise to non-identity fixed points for the shift.
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