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, a 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 following reading path moves 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 the concepts through the theorem, method, and distribution indexes. Mixture and variance-gamma theory links to the existing Normix notes.
Maximum entropy derives a Pareto model from a fixed logarithmic moment, with explicit support and reference measure.
Choose an entry path¶
For an introduction with basic calculus and probability, read Pareto, moment existence, running means, and max-to-sum ratios, then regular variation. These pages separate finite moments from reliable finite-sample summaries.
To analyze data:
Begin with the NYC dog-name example. Specify the observed unit and which tail is being studied.
Compare empirical survival with candidate models, then use threshold selection. Body/shoulder geometry concerns mixture densities; its curvature boundaries do not select empirical thresholds.
Compare Hill and mean excess, reporting threshold sensitivity and the example’s discrete-count and dependence caveats.
The S&P method extends the workflow to daily losses when an authorized local data cache is available.
Evidence and exposure¶
Inference asymmetry constructs two laws with close finite-sample behavior and very different means.
Asymmetric payoffs separates gain frequency, expected payoff and the effect of curvature.
Ruin gives finite-horizon survival formulas and explains why a growing expectation can coexist with terminating paths.
Decentralized exposure distinguishes nonlinear damage reduction from independence between risks.
Mixtures and moment estimation¶
Variance mixtures hold variance fixed while changing kurtosis, or change the moment regime by mixing precision.
Body and shoulder geometry separates variance curvature, scale curvature, and finite-mixture crossings.
Moment estimation distinguishes a population moment from the moments and consistency of its estimator.
Dependence in several dimensions¶
Multivariate tails builds a Student vector from a shared scale, derives its mutual information, and separates finite sample statistics from population moments. Density geometry compares identical marginals under different joint laws.
Reading and reference¶
Empirical examples: two committed datasets, two simulations, and the S&P method with explicit data availability.
SCoFT guide: Chapter 1 — Prologue, Chapter 2 — Glossary, definitions, and notations, Chapter 3 — Non-technical overview, Chapter 4 — Finite-moment fat tails, Chapter 5 — Subexponentials and power laws, and Chapter 6 — Thick tails in higher dimensions. Chapter 7 remains planned.
External guides, including Embrechts, Klüppelberg, and Mikosch.
Concept graph, with a plain linked fallback.
Figures and calculations are static. Reproduction scripts run separately from the site build; neither notebooks nor data fetches execute during publication.
The mixture sample introduces a complementary question: how conditioning specifies a model rather than counts observations.