What Is a Heavily-Tailed Distribution?

A heavily-tailed distribution, also written as a heavily tailed distribution, is a probability distribution whose tail decays more slowly than any exponential function, so that extreme values occur far more often than a normal or exponential model would predict. The formal condition is that the moment generating function is infinite for every positive argument: no exponential bound of the form exp(-λx) dominates the tail probability as x grows. Distributions in this class include the Pareto, log-normal, Weibull with shape parameter less than one, Cauchy, and the stable laws with index below two. Their defining practical consequence is that the largest observation in a sample is often comparable in size to the sum of all the others, so averages converge slowly and sample variance may not converge at all.

The concept arose independently in several fields: in actuarial mathematics, where a single catastrophic claim dominates a portfolio, in economics through Vilfredo Pareto's work on income, and in communications engineering, where measured packet arrivals and file sizes proved incompatible with the Poisson and exponential models inherited from telephony.

Subclasses and Formal Definitions

Within the heavy-tailed family, engineers and statisticians work mainly with two nested subclasses. A distribution is regularly varying with index α when the tail probability behaves like a slowly varying function multiplied by x raised to the power of negative α, which is the precise statement of a power law tail. A distribution is subexponential when the probability that a sum of two independent copies exceeds a large threshold is asymptotically twice the probability that one copy does, a property usually summarized as "the sum behaves like the maximum." Every regularly varying distribution is subexponential, and every subexponential distribution is long tailed and therefore heavy tailed, but the converses fail. Treatments such as the monograph on heavy-tailed and subexponential distributions by Foss, Korshunov, and Zachary set out these implications and the counterexamples that separate the classes.

Estimating and Testing Tail Behavior

Fitting a heavy tail to data is harder than fitting a central tendency, because only the largest observations carry information about the tail index and there are by construction few of them. The classical tool is the Hill estimator, which uses the k largest order statistics, but its output depends strongly on the choice of k and can produce a plausible looking index for data that follow no power law at all. Log-log plots of the empirical survival function are similarly unreliable, since curvature is easy to miss by eye. The now standard procedure, set out in the widely cited treatment of power-law distributions in empirical data, combines maximum likelihood estimation of the exponent with a data-driven choice of the lower cutoff, a Kolmogorov-Smirnov goodness-of-fit test based on synthetic resampling, and likelihood ratio comparison against competing models such as the log-normal.

Mechanisms and Modeling Consequences

Heavy tails arise from identifiable generative mechanisms rather than by accident. Multiplicative growth with a reflecting barrier, preferential attachment in growing networks, random recursion equations, and mixtures of exponentials with a heavy-tailed rate parameter all produce power law behavior, and the survey of heavy-tailed distributions in stochastic dynamical models collects these routes in a single framework. The modeling consequences are severe. Queues fed by heavy-tailed service times exhibit waiting times far longer than Poisson analysis predicts, which is why self-similar traffic models replaced Markovian ones in network capacity planning. Risk measures based on variance become meaningless when the second moment diverges, and confidence intervals built on the central limit theorem converge so slowly that they misstate uncertainty at realistic sample sizes.

Applications

Heavily-tailed distributions have applications in a wide range of disciplines, including:

• Network traffic modeling, buffer sizing, and queueing analysis
• Insurance and reinsurance pricing for catastrophic loss
• Quantitative finance, in value-at-risk and portfolio tail risk estimation
• Reliability engineering and extreme value analysis of component failure
• Analysis of degree distributions in social, biological, and technological networks
• Hydrology and seismology, for flood and earthquake magnitude recurrence