States of randomness from mild to wild, and concentration from the short to the long run

Abstract

An innovative useful metaphor is put forward in this chapter, and described in several increasingly technical stages. Section 1 is informal, but Sections 4 and 5 are specialized beyond the concerns of most readers; in fact, the mathematical results they use are new.

At the core is a careful examination of three well-known distributions: the Gaussian, the lognormal and the scaling with infinite variance (α < 2). They differ deeply from one another from the viewpoint of the addition of independent addends in small or large numbers, and this chapter proposes to view them as “prototypes,” respectively, of three distinct “states of randomness:” mild, slow and wild. Slow randomness is a complex intermediate state between two states of greater simplicity. It too splits more finely, and there are probability distributions beyond the wild.

Given N addends, portioning concerns the relative contribution of the addends U_n to their sum $\sum\nolimits_1^N {{U_n}}$. Mildness and wildness are defined by criteria that distinguish between even portioning, meaning that the addends are roughly equal, ex-post, and concentrated portioning, meaning that one or a “few” of the addends predominate, ex-post. This issue is especially important in the case of dependent random variables (Chapter E6), but this chapter makes a start by tackling the simplest circumstances: it deals with independent and identically distributed addends.

Classical mathematical arguments concerning the long-run (N→∞) will suffice to distinguish between the “wild” state of randomness and the remaining states, jointly called “preGaussian.”

Novel mathematical arguments will be needed to tackle the short-run (N = 2 or “a few”). The resulting criterion will be used to distinguish between a “mild” or “tail-mixing” state of randomness, and the remaining states, jointly called “long-tailed” or “tail-preserving.” This discussion of long-tailedness may be of interest even to readers reluctant to follow me in describing the levels of randomness as “states.”

In short-run partition, short-run concentration will be defined in two ways. The criterion needed for “concentration in mode” will involve the convexity of log p(u),where p(u) is the probability density of the addends. The concept of “concentration in probability” is more meaningful but more delicate, and will involve a limit theorem of a new kind. Long-tailed distributions will be defined by the very important “tail-preservation criterion” under addition; it is written in shorthand as P _N ~ NP.

Randomness that is “preGaussian” but “tail-preserving” will be called “slow.” Its study depends heavily on middle-run arguments (N = “many”) that involve delicate transients.