标题： 具有指数增长样本空间的过程的概率和信息熵的特征及其在广义货币供应量中的应用 显示英文标题

标题： Characterization of the probability and information entropy of a process with an exponentially increasing sample space and its application to the Broad Money Supply

摘要： 存在一个随机变量（X），其结果是确定的（即，X = x0），p(x0) = 1。 考虑x0在整数区间[1, s]上具有离散均匀分布，在初始状态下样本空间大小(s) = 1，使得p(x0) = 1。 当s以指数方式增加时，x0的概率及其相关的信息熵(H)是多少？ 如果样本空间以指数速率（速率常数= λ）随时间(t)扩展，并应用时间尺度变换，使得T = λ × t，则有：p(x0|T) = exp(-T)和 H(T) = T。 该特征描述还通过同时独立过程以及更一般的多指数情况扩展到包含指数扩展。 该方法应用于2001年至2019年期间美国广义货币供应量的增长，作为一个现实世界的例子。 在任何给定时刻，信息熵与样本空间扩展的速度有关。 在广义货币供应扩展的背景下，信息熵可以被认为与货币供应扩展的“速度”有关。 显示英文摘要

摘要： 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 exponentially? If the sample space expansion occurs at an exponential rate (rate constant = lambda) with time (t) and applying time scaling, such that T = lambda x t, gives: p(x0|T)=exp(-T) and H(T)=T. The characterization has also been extended to include exponential expansion by means of simultaneous, independent processes, as well as the more general multi-exponential case. The methodology was applied to the expansion of the broad money supply of US$ over the period 2001-2019, as a real-world example. 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.