Tail Class Catalog

Main facts¶

The SCoFT Prologue guide separates Gaussian-relative kurtosis from asymptotic tail classes. This catalog compares nonnegative right-tail models by subexponentiality and regular variation, then checks which moments exist. The definitions concern a specified population law.

The positive classifications are cited, not proved here: Foss, Korshunov, and Zachary, second edition (2013), §3.5, Theorem 3.29 and its following paragraph, p. 57 (regular variation, including Pareto); Theorem 3.30 and its following examples, p. 58 (lognormal and Weibull with shape below one). See the book. We derive the elementary tail ratios and moment contrasts below; the appendix proves the convolution decomposition.

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. See Shared notation.

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 $[0,\infty)$. The converse fails: lognormal and stretched-Weibull tails are subexponential without being regularly varying.

Slow variation need not settle to a constant¶

SCoFT v4 §§2.2.1 and 2.2.11, pp. 9 and 13, discuss a practical transition to a nearly constant tail factor Taleb, 2025. Such a transition is an extra modeling assumption. The definition of slow variation does not imply a finite constant limit, or an exact Pareto tail beyond a finite threshold.

Here is a complete counterexample, in dimensionless units:

We prove that this is a survival function with a regularly varying tail. It is continuous at 1, tends to zero at infinity, and its derivative on $x>1$ is $-x^{-3}(1+2\log x)<0$. Thus it defines a probability law on $[1,\infty)$. Its factor $L(x)=1+\log x$ tends to infinity, while for every fixed $t>0$ and sufficiently large $x$,

The chapter 2 worked cases evaluate this ratio alongside the exact Pareto ratio $1/4$ at multiplier 2. A finite curve cannot establish where a tail approximation becomes adequate; threshold selection requires its own diagnostics. The example disproves a constant-limit requirement analytically.

Moment gates are different questions¶

The Cramér condition asks whether $\mathbb E e^{\theta X}<\infty$ for at least one $\theta>0$. Ordinary moments ask about $\mathbb E X^q$, with $q>0$. For the following nonnegative models:

The Pareto moment proof gives the last row’s power-moment boundary, including divergence at $q=\alpha$; the Cramér note proves its exponential-moment failure. For the exponential row, direct integration gives $\int_0^\infty x^q e^{-x}\,dx=\Gamma(q+1)<\infty$ and $\int_0^\infty e^{-(1-\theta)x}\,dx=1/(1-\theta)$ when $\theta<1$; the latter integral diverges when $\theta\ge1$. Here $\Gamma$ denotes the gamma integral.

Proof of the lognormal moment contrast¶

We prove both entries in the lognormal row. Let $Z\sim N(0,1)$ and $X=e^Z$. Completing the square gives, for every real $q$,

In contrast, for any fixed $\theta>0$,

Indeed, $e^z/z^2\to\infty$ as $z\to\infty$, so the integrand is at least 1 for all sufficiently large $z$. The integral over that half-line diverges. This establishes failure of the Cramér condition despite all positive power moments being finite. Subexponentiality is a separate, cited result; it does not follow just from this failure.

For a Pareto model with $\alpha=5/2$ and $x_m=1$, the mean is $5/3$ and the variance is $20/9$, but the fourth moment diverges. The formulas follow from the linked integral: $\mathbb E X^2=5$ and $5-(5/3)^2=20/9$. Pearson kurtosis is therefore not a finite summary for this model. Having a finite variance does not imply a finite fourth or positive exponential moment.

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, Eq. 7.12.1. It also gives $\log\bar F(x)=-\tfrac12(\log x)^2-\log\log x-\tfrac12\log(2\pi)+o(1)$ for this standard lognormal as $x\to\infty$. Adding $cx$ or $p\log x$ proves the two exponential-versus-power comparisons stated above. For $\log X\sim N(\mu,\sigma^2)$ with $\sigma>0$, replace $\log x$ in the normal-tail calculation by $(\log x-\mu)/\sigma$; the same comparisons and zero multiplier-ratio limit follow. 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¶

We prove the two negative classifications in the table. Write $\operatorname{Gamma}(k,\lambda)$ in shape–rate notation, with $k,\lambda>0$. At rate one, convolution of the densities with shapes $k$ and $\ell>0$ gives

using the beta integral (NIST DLMF, Eq. 5.12.1). Thus independent shapes add. Scaling by $1/\lambda$ gives the same rule at a common rate $\lambda$. The survival integral, compared with its leading integrand by l’Hôpital’s rule, gives

For completeness, at rate one the derivatives of $\int_x^\infty y^{k-1}e^{-y}\,dy$ and $x^{k-1}e^{-x}$ have ratio $[1-(k-1)/x]^{-1}\to1$; both functions tend to zero. Consequently, for fixed $t>1$,

This proves that every positive-shape gamma law, including the exponential case $k=1$, is neither regularly varying with a finite positive exponent nor subexponential.

For integer $k\ge1$ and rate one, repeated integration by parts gives the exact survival $e^{-x}\sum_{j=0}^{k-1}x^j/j!$ for $x\ge0$. Hence

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.

A finite model comparison¶

Use pareto_type1, log_survival_multiplier_ratio.

from scipy.stats import expon, lognorm
from xmath.incerto.distributions import pareto_type1
from xmath.incerto.tail_classes import log_survival_multiplier_ratio

for label, law in (("Pareto", pareto_type1(2.5)),
                   ("Lognormal", lognorm(1)), ("Exponential", expon())):
    log_ratio = log_survival_multiplier_ratio(law.logsf, [2, 10, 100])
    print(f"{label}: [" + ", ".join(f"{v:.4g}" for v in log_ratio) + "]")

Output

Pareto: [-1.733, -1.733, -1.733]
Lognormal: [-1.081, -2.052, -3.563]
Exponential: [-2, -10, -100]

A finite grid illustrates the specified laws and cannot classify unknown data.

Figure regeneration: uv run python demos/incerto/prologue_tail_classes.py from the repository root. It evaluates $\log A_2(u)$ for Pareto $(\alpha=5/2,x_m=1)$, standard lognormal, Weibull $(\beta=1/2)$, exponential (rate 1), and gamma (shape 2, rate 1). The fixed grid has 241 logarithmically spaced thresholds from 1 to 10⁴; all quantities are dimensionless population calculations. There is no sampling, data fitting, truncation of a distribution, or uncertainty interval.

The script also prints exact exponential and gamma two-sum checks at $u=2,10,100$. The helper uses log survival values for the multiplier figure; non-finite values cause an error. In particular, it does not divide zeros when an ordinary survival function underflows. The figure is numerical illustration of the formulas above, not evidence of an observed tail class.

A survival that varies slowly itself¶

The positive-exponent convention in the table does not exhaust regularly varying functions. To examine the index-zero boundary, let $V$ be Pareto with minimum $v_m>0$, exponent $a>0$, and dimensionless values, and put $X=e^V$. Here $a,v_m$ are local parameters; the survival exponent of $X$ will be zero. This construction is motivated by SCoFT v4 §5.5, pp. 102–103 Taleb, 2025. Inverting the increasing transform proves

For every fixed $t>0$, its survival ratio tends to one because $\log(tx)/\log x\to1$. Thus $\bar F_X\in RV_0$, outside this catalog’s positive-power rows. Every positive power moment diverges: $\mathbb EX^p=\mathbb E e^{pV}$, and the exponential factor eventually exceeds every polynomial in the Pareto density integral.

We also prove subexponentiality directly for this example. For iid nonnegative $X_1,X_2$, with $M_2=\max(X_1,X_2)$,

Both bounds divided by $2\bar F_X(x)$ tend to one: the maximum-tail formula is $2\bar F_X-\bar F_X^2$ and $\bar F_X(x/2)/\bar F_X(x)\to1$. The squeeze proves the claim. Thus a regularly varying survival can be subexponential even when its index is zero. There is no finite positive $1/\alpha$ calibration at this index-zero boundary.

The chapter demo evaluates this survival using $\log x=2,4,8$ with $a=2$, $v_m=1$, obtaining $1/4,1/16,1/64$. It keeps the thresholds in logarithmic coordinates to avoid overflowing $e^V$. Finite evaluations do not stand in for the moment-divergence or convolution proofs.

The maximum-entropy note gives another construction of power tails, conditional on specified information constraints. Neither construction establishes a tail law for observed data.

Regime separation is not a tail class¶

Appendix A of SCoFT v4, §A.1, pp. 122–124 Taleb, 2025, considers distributions with separated regimes. We can check the effect on kurtosis without assuming a power tail. Let $B$ take -1 and 1 with equal probabilities, let $Z\sim N(0,1)$ be independent, and set $X=mB+\sigma Z$, with $m\ge0$ and $\sigma>0$. Symmetry gives mean zero. Expanding and discarding the odd terms proves

Here $K$ is Pearson kurtosis. For $m>0$ it lies below three and tends to one as $m/\sigma\to\infty$. Yet $\mathbb E e^{tX}=\cosh(mt)e^{\sigma^2t^2/2}<\infty$ for every real $t$. This location mixture is light-tailed. Its density has a local minimum at zero when $m>\sigma$: differentiating the two normal densities gives $f''(0)=f(0)(m^2-\sigma^2)/\sigma^4>0$. Thus low kurtosis can accompany a valley around the expectation. It is not evidence of a small probability of a large regime change.

For $m=2$, $\sigma=1$, the variance is 5 and $K=43/25=1.72$. The variance-mixture note holds the location fixed and mixes variances; these are different experiments. A finite normal mixture has positive density everywhere, so a valley is not a forbidden interval and every individual point still has probability zero.

An absorbing state needs repeated chances to be reached¶

For the §A.2 context, pp. 124–125, take a homogeneous two-state Markov chain with states intact and failed. The failed state is absorbing and an intact system fails with probability $p\in[0,1]$ at each step, conditional on still being intact. Its row-stochastic transition matrix is

Induction proves the power: multiplying by $T$ multiplies the remaining intact probability by $1-p$. Starting intact, eventual failure has probability one if $p>0$ and zero if $p=0$. Irreversibility alone is not sufficient; reachability and persistent conditional risk matter. With varying conditional hazards $p_j$, intact survival is $\prod_{j=1}^n(1-p_j)$. For example, $p_j=2^{-j-1}$ has sum $1/2$, and $\prod_{j=1}^n(1-p_j)\ge1-\sum_{j=1}^n p_j\ge1/2$ by induction. Every step has positive risk but eventual failure is not certain. The existing survival calculation treats this product; we do not replace it with an independence assumption on unconditional events.

Use normal_location_mixture_kurtosis, survival_from_hazards.

from xmath.incerto.multivariate import normal_location_mixture_kurtosis
from xmath.incerto.lecture_risk import survival_from_hazards

print(f"Location-mixture kurtosis: {normal_location_mixture_kurtosis(2):.4g}")
print(f"Constant hazard: {survival_from_hazards([0.05] * 100)[-1]:.4g}")
hazards = [2.0**(-j - 1) for j in range(1, 101)]
print(f"Decreasing hazard: {survival_from_hazards(hazards)[-1]:.4g}")

Output

Location-mixture kurtosis: 1.72
Constant hazard: 0.005921
Decreasing hazard: 0.5776

Neither example asserts a power-tail law.

The Pareto mixture contrast shows what changes when one regime actually has a power tail.

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¶

• Taleb (2025), appendix A, §A.1, pp. 122–124, and §A.2, pp. 124–125: regime-mixture and transition context. The symmetric-location and absorbing-chain special cases are proved above.

• Taleb (2025), §5.0.1, pp. 91–93, and §5.5, pp. 102–103: classification and log-Pareto context. The new index-zero calculation is proved here; earlier positive classifications retain the references below.

• Bingham et al. (1987), Regular Variation.

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

• Foss, Korshunov, and Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions 2013, 2nd ed., §3.5, Theorem 3.29 and following paragraph, p. 57; Theorem 3.30 and following examples, p. 58. These support the cited positive classifications.

• Taleb, Statistical Consequences of Fat Tails, third edition (2025), arXiv:2001.10488v4, ch. 1, terminology discussion, pp. 4–5: context for the companion Prologue guide, not a theorem source for this table.

Backlinks¶

• Depends on: Subexponentiality and Regular Variation.

• Example distribution: Pareto Distribution.

• Related diagnostic: Max-to-Sum Ratio.