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Characterization of the probability and information entropy of a process with an increasing sample space by different functional forms of expansion, with an application to hyperinflation

There is a random variable (X) with a determined outcome (i.e., X = x0), p(x0) = 1. Consider x0 to have a discrete uniform distribution over the integer interval [1, s], where the size of the sample space (s) = 1, in the initial state, such that p(x0) = 1. What is the probability of x0 and the associated information entropy (H), as s increases by means of different functional forms of expansion? Such a process has been characterised in the case of (1) a mono-exponential expansion of the sample space; (2) a power function expansion; (3) double exponential expansion. The double exponential expansion of the sample space with time (from a natural log relationship between t and n) describes a "hyperinflationary" process. Over the period from the middle of 1920 to the end of 1923, the purchasing power of the Weimar Republic paper Mark to purchase one gold Mark became close to zero (1 paper Mark = 10 to the power of -12 gold Mark). From the purchasing power of the paper Mark to purchase one gold Mark, the information entropy of this hyperinflationary process was determined.

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