## Matthew Nicol, University of Houston

Statistical properties of dynamical systems

**
Thursday October 13, 4:00PM,
Herman Brown 227**
Sequences of independent, identically distributed, (iid) random variables subject to mild moment
conditions satisfy the central limit theorem
(CLT) (convergence in distribution of the
normalized averages to a normal distribution), the law of the iterated logarithm (LIL) (which
quantifies, almost surely, the largest deviation of the normalized averages from the mean), and the
almost sure invariance principle (ASIP).
The ASIP is an almost sure approximation
of partial sums by Brownian motion which implies the CLT and the LIL. Time-series of observations on
a dynamical system may exhibit the same statistics. We will survey results on statistical
properties of
dynamical systems, such as the CLT and ASIP.
We then describe recent work on
establishing the ASIP for Hölder observations on a broad class of non-uniformly hyperbolic
dynamical systems.