Speaker

Laurence Francis Lacey, Director of Lacey Solutions Ltd, Dublin, Ireland.

Abstract

There is a random variable (X) with a determined outcome (i.e., \(X = x_0\)), \(p(x_0) = 1\). Consider \(x_0\) 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 \(p(x_0) = 1\). What is the probability of \(x_0\) and the associated information entropy (\(H\)), as \(s\) increases by expansion of different functional forms? Such a process has been characterised in the case of (1) a monoexponential expansion of the sample space; (2) a power function expansion; (3) double exponential expansion. What determines the different functional forms of the expansion of the sample space with time is the functional relationship between time and \(n\) (no. of sample space doublings). If the expansion of the sample space occurred simultaneously via three (or more) independent processes, of different functional forms, the combined overall process can also be determined.

The methodology was applied to the expansion of the broad money supply of $US over the period 2001-2019, as a real-world example of the mono-exponential expansion of the sample space. At any given time, the information entropy is related to the rate at which the sample space is expanding. In the context of the expansion of the broad money supply, the information entropy could be considered to be related to the “velocity” of the expansion of the money supply.

The double exponential expansion of the sample space with time describes a “hyperinflationary” process. In terms of the fiat money supply, such an expansion would lead to hyperinflation such that the “money” would become increasingly “worthless” with time. 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 = \(10^{-12}\) gold Mark). From the purchasing power of the paper Mark to purchase one gold Mark, the information entropy of this hyperinflationary process was determined.

References

Lacey LF (2021). Characterization of the probability and information entropy of a process with an exponentially increasing sample space and its application to the Broad Money Supply. See here.

Lacey LF (2021). 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. See here.

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