Tail questions begin by specifying an event and the evidence available for it. A fraction counted in a finite sample and a probability under a proposed model are different quantities. Keeping both visible makes later estimation arguments easier to assess.
Start with counting exceedances, an original worked example with a short Python demo. It uses only a finite list and strict inequalities; no moment or distributional assumption is needed for the count.
The first imported concept batch develops a path from a distributional model to estimates of tail quantities:
Pareto distribution specifies an exact power tail; regular variation allows slowly varying corrections at high thresholds.
Karamata’s theorem connects tail shape to integrals. Pareto moment existence proves the exact moment boundaries, and the mean excess function describes overshoots beyond a threshold.
The Hill estimator estimates a positive tail index from log-excesses. The extreme-value index places that estimate within the wider classification of extreme-value limits.
The generalized Pareto distribution models threshold excesses, and plug-in tail estimation explains how a fitted tail model enters a mean estimate, including its moment-boundary singularity and the contribution of the distribution body.
The second reading path connects extreme-value limits to diagnostics.
Peaks over threshold & maxima¶
The Pickands–Balkema–de Haan theorem justifies the GPD excess model under a maximum-domain-of-attraction assumption.
The generalized extreme-value distribution describes normalized block-maxima limits; the Frechet law is its positive-shape member and connects those limits to regularly varying tails.
Tail threshold selection makes the finite-sample tuning decision explicit through exceedance counts, shape, mean excess, and modified-scale stability.
One-big-jump diagnostics¶
Subexponentiality explains why a large sum of a fixed number of nonnegative iid terms has the same leading tail as its maximum.
The survival tail ratio compares two moderate extremes with one doubled extreme. The max-to-sum ratio measures the largest observation’s share of a realized total. Their limits answer different questions and neither diagnostic alone proves an empirical tail class.
Symmetric shifted double Pareto provides a two-sided power-tail model, with a clear distinction between symmetry and existence of the mean.
Body, shoulders, and tails derives local density changes under variance mixing. Its curvature boundaries describe mixture geometry; choosing a Pareto threshold remains a separate task.
Iso-density tail geometry compares equal and axial splits of a large sum under normal and Cauchy densities. Density at a point and integrated tail probability remain distinct.
The third reading path contrasts light-tailed concentration with stable sums and infinite-mean averages.
Sums, stable limits, and thin-tail contrast¶
The Cramér exponential-moment condition gives a Chernoff bound and separates positive exponential moments from ordinary moment existence.
The generalized central limit theorem describes stable limits of normalized iid sums, including tail balance and centering at the index-one boundary.
LLN failure under infinite mean proves that averages of nonnegative iid variables with infinite mean diverge almost surely. Capped Pareto means make the truncation argument explicit.
Tail catalog & indexes¶
The tail class catalog compares regular variation and subexponentiality for Pareto, lognormal, Weibull, exponential, and gamma examples. Browse all imported concepts through the theorem, method, and distribution indexes. Mixture and variance-gamma theory links to the existing Normix notes.
These twenty-six concept imports comprise twenty-three body notes and three indexes, alongside the original counting-exceedances example and this track hub. The notes retain mathematical arguments and use static calculations; they do not execute notebooks during the site build. Use the shared notation canon for overlapping symbols. Data examples, reading guides, and dependency-graph navigation remain planned.
The mixture sample introduces a complementary question: how conditioning specifies a model rather than counts observations.