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Sigma Series in Stochastics, Volume 1, 147--174
Heldermann Verlag 2004

Theoretical Stochastics

Elart von Collani
University of Würzburg, Sanderring 2, 97070 Würzburg, Germany

Stochastics as an independent branch of natural science can be based neither on probability theory nor on statistics, but needs its own theoretical fundaments. These are developed in the present paper. Uncertainty refers to the change from past to future and, therefore, variables are necessary describing the determinate past and the indeterminate future development. Moreover, functions are needed which model the range of variability and the structure of variability and thus the laws of change. The paper introduces the necessary quantities and develops generic representations for them. Moreover, classifying the structure of randomness leads to the definition of disjoint families of probability distributions allowing to assign an appropriate type of probability distribution to a given random variable with known qualitative behavior.