Generalized Bernoulli Process and Fractional Binomial Distribution
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Publication:5984476
DOI10.48550/ARXIV.2209.01516arXiv2209.01516MaRDI QIDQ5984476
Author name not available (Why is that?)
Publication date: 4 September 2022
Abstract: We propose a generalized Bernoulli process that can approximate the fractional Poisson process. A Bernoulli process is a sequence of independent binary random variables, and the binomial distribution concerns the behavior of the sum of the variables in a Bernoulli process. Recently, a generalized Bernoulli process (GBP-I) was developed in which binary random variables are correlated with its covariance function obeying a power law. In this paper, we propose a different generalized Bernoulli process (GBP-II) that can possess long-memory property as the GBP-I but has a different limiting behavior. We show that the sum of the variables in the GBP-I can serve as an over-dispersed binomial model when long-memory presents, whereas the GBP-II can be considered as a discrete-time counterpart of the fractional Poisson process. Considering that the binomial distribution approximates the Poisson distribution under certain conditions, the connection we found between the GBP-II and the fractional Poisson process is thought of as its counterpart under long-range dependence. Data analysis with the unemployment rate shows that under long-range dependence, the GBPs fit the data better than a Markov chain.
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