Beyond the law of large numbers: Introducing progressive sampling, weaving, the geometric triangle, and corresponding distributions

TL;DR

This paper introduces progressive sampling and weaving as novel probabilistic models that challenge traditional convergence assumptions, revealing new distributions and fractal structures through an analytically tractable framework.

Contribution

It presents a new model of sampling that avoids convergence, introduces weaving, and uncovers associated distributions and fractal structures, expanding probabilistic theory.

Findings

Discovery of a multiplicate structure akin to Pascal's triangle

Introduction of a new class of distributions related to the binomial

Identification of a fractal structure emerging from the probabilistic model

Abstract

In probability theory and statistics, the IID model represents a single population, and a large, potentially infinite sample from this population. Main theorems, in particular the central limit theorem and laws of large number (LLN) assure convergence, making asymptotic statistics possible. To avoid convergence, it is thus straightforward to consider two populations and a sample that ceaselessly fluctuates between them. It is the aim of this contribution to study the effects that thus occur. To this end, we introduce "progressive sampling," leading to a straightforward model that is analytically tractable. With a minimum of technical overhead, a number of interesting results thus ensue: In particular, one encounters a multiplicate structure (similar to Pascal's triangle) that is associated with a new class of distributions (related to the binomial). Although the argument is completely probabilistic, it entails a well-known fractal structure. It also turns out that the new (global) operation of "weaving" is equivalent to a certain (local) cascade process.