Tail Class Catalog

Main facts¶

We catalog common right-tail examples by two asymptotic properties: subexponentiality and regular variation. The classifications are cited results; the appendix proves the elementary convolution decomposition. Standard references are Bingham, Goldie, and Teugels for regular variation 1987, and Embrechts, Klueppelberg, and Mikosch 1997 plus Foss, Korshunov, and Zachary 2013, 2nd ed. for subexponential tails.

Let $\bar F(x)=\mathbb P(X>x)$ be the survival function. In the table, “regularly varying” means $\bar F\in RV_{-\alpha}$ for some $\alpha>0$, so $\bar F(tx)/\bar F(x)\to t^{-\alpha}$ for fixed $t>0$. Here $F$ is the CDF, $x$ is the threshold, and $\alpha$ is a positive tail exponent. All examples below have nonnegative support, with positive scale parameters, positive gamma shape and rate, and nonzero lognormal log-variance. Shared notation is planned.

For the lognormal row, “heavier than any exponential, lighter than any power” means the survival function satisfies

The inclusion direction to remember is:

for distributions supported on $[0,\infty)$, or under the standard corresponding right-tail assumptions. The converse fails: lognormal and stretched-Weibull tails are subexponential without being regularly varying.

Reusable diagnostics¶

For a nonnegative continuous law with unbounded support, compare

where $X_1,X_2$ are iid and $t>1$ is fixed. The first ratio checks multiplicative tail scaling; the second is the defining two-summand subexponential ratio. These are population quantities. Finite evaluations of them do not certify an asymptotic class from data.

Here $\bar\Phi$ is the standard normal survival function. For the lognormal row, the normal-tail asymptotic $\bar\Phi(z)\sim\phi(z)/z$, with $\phi$ the standard normal density, yields

The asymptotic for $\bar\Phi$ follows from the complementary-error-function expansion in NIST DLMF, Section 7.12. Thus the multiplier diagnostic separates power tails from all four other examples, but cannot distinguish subexponential lognormal and stretched-Weibull tails from the two light-tailed examples. The classifications of lognormal and stretched Weibull are cited results; a ratio table is not a proof of their subexponentiality.

Exact light-tail checks¶

Exponential and integer-shape gamma sums give elementary checks of $R$. With shape–rate notation, $\operatorname{Gamma}(k,1)$ has survival $e^{-x}\sum_{j=0}^{k-1}x^j/j!$ for $x\ge0$. Independence adds the shapes. Consequently,

Both ratios diverge, although the gamma ratio happens to be near one at $x=2$. A single moderate-threshold observation is insufficient to infer a limiting tail class. The Cramér condition provides the exponential-moment contrast for these two light-tailed laws.

Caveats¶

• Regular variation and subexponentiality concern limits, not the shape of a fitted curve on a finite range.

• Very small survival probabilities can underflow in numerical calculations. Logarithmic ratios help preserve information; rounded zeros do not establish exact zero probability.

• The subexponential definition used here is the right-tail iid version for nonnegative summands. Two-sided and dependent settings need additional assumptions.

• The finite exact examples above check formulas, not the accuracy of an upstream sampler, survival-function implementation, or quadrature routine.

Appendix: convolution calculation¶

This appendix proves the elementary two-summand convolution decomposition used by the subexponential diagnostic. We work with independent continuous variables, then specialize to an iid lower-bounded distribution. Discrete, dependent, and two-sided variants require separate assumptions.

For independent continuous random variables $X_1$ and $X_2$ with densities $f_1$, $f_2$ and survival functions $\bar F_1$, $\bar F_2$,

To see the iid reduction, take a common density $f$, survival function $\bar F$, and lower support endpoint $a$, with $x>2a$. Split the event $\{X_1+X_2>x\}$ into three cases, ignoring probability-zero boundary points: first $X_2\le x/2$, second $X_1\le x/2$, and third both variables exceed $x/2$. In the first case, conditioning on $X_2=y$ with $a\le y\le x/2$ leaves the requirement $X_1>x-y$, so this part contributes $\int_a^{x/2} f(y)\bar F(x-y)\,dy$. The second case contributes the same quantity by iid symmetry. The remaining upper-right square has probability $\bar F(x/2)^2$ by independence.

For $a=0$, the partition can be recorded as disjoint regions rather than a contour plot. Boundaries have zero probability under the continuous-law assumption.

For iid variables with lower support endpoint $a$, symmetry gives the more useful half-line formula

Dividing by $2\bar F(x)$ yields a normalized integral:

The logarithmic form applies where the factors are positive, with zero terms understood by limits. It avoids forming tiny unnormalized probabilities before taking their ratio. The decomposition is an ordinary conditioning argument; no numerical integration or executable package helper is needed for it.

References¶

• Bingham, Goldie, and Teugels, Regular Variation 1987.

• Embrechts, Klueppelberg, and Mikosch, Modelling Extremal Events 1997.

• Foss, Korshunov, and Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions 2013, 2nd ed..

Backlinks¶

• Depends on: Subexponentiality and Regular Variation.

• Example distribution: Pareto Distribution.

• Related diagnostic: Max-to-Sum Ratio.

Source and adaptation¶

Adapted from incerto-wiki, content/concepts/distributions/tail-class-catalog.md, revision 9717c9c (2026-09-13 import). Copyright (c) 2023 xshi19. Licensed under MIT. Links and notation were adapted for this site; executable figures and simulations were replaced with static calculations. No upstream execution or formal-proof verification is claimed for this adaptation.