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

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.