Berlin 2012 – wissenschaftliches Programm

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DY: Fachverband Dynamik und Statistische Physik

DY 2: Statistical Physics (General)

DY 2.7: Vortrag

Montag, 26. März 2012, 11:30–11:45, MA 004

The Central Limit Theorem in Hierarchical Structures — •René Pfitzner, Pavlin Mavrodiev, Ingo Scholtes, Claudio J. Tessone, and Frank Schweitzer — Chair of Systems Design, ETH Zurich, Switzerland

In statistics the (classical) Central Limit Theorem for independent and identically distributed variables is well known. A generalization of this theorem is the so-called Lyapunov Central Limit Theorem, which is applicable to settings with independent, but not necessarily identically distributed random variables. In this contribution, we generalize these theorems in a hierarchical setting i.e., the aggregation of random variables is performed in a step-wise fashion where first sub-groups of all variables get aggregated. We show that the non-linearity introduced by the hierarchical organization of the variables leads to an interesting effect: not all the aggregation schemes lead to the same variance at the end of the hierarchy. In fact, there is an optimum hierarchical structure that minimizes the final error, or variance. We pose an optimization problem to find the "most-beneficial" hierarchical scheme for aggregation. We argue that our results have broad implications, ranging from the arrangement of measurements taken with devices with different intrinsic precision, to group decision making.