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Extreme value theory studies limiting probability distributions of normalized sample maxima and related extremes. The standard limits for maxima include the Fréchet distribution, negative Weibull distribution, and Gumbel distribution.

The question is often practical: if a batch finishes only when its slowest task finishes, its completion time is a sample maximum. A typical individual delay or a sample mean does not answer that question. The central limit theorem describes suitably normalized sums and sample means under its assumptions; it does not give a limit law for sample maxima. Extreme value theory instead describes how the largest observation changes as the number of opportunities for a large observation grows.

For independent and identically distributed random variables $X_1,X_2,\dots$ with distribution function $F$, define the sample maximumThe event $M_n\le x$ means that every observation is at most $x$, so independence givesExtreme value theory asks whether there are constants $a_n>0$ and $c_n\in \mathbb{R}$ for which $(M_n-c_n)/a_n$ has convergence in distribution to a non-degenerate probability distribution. Equivalently, $F(c_n+a_{n}z{)}^n$ must tend to a limiting distribution function at its continuity points. The extremal types theorem says that any such limit belongs, up to a change of location and positive scale, to the Gumbel distribution, Fréchet distribution, or negative Weibull distribution. It does not guarantee that a limit exists for every $F$. ^[ref].

The maximum of independent exponential random variables gives a particularly transparent worked example: its center grows as $(\log n)/\lambda$, and its fluctuations have a Gumbel distribution limit on the scale $1/\lambda$. Here and below $\log$ denotes the natural logarithm. This normalization depends on the exponential distribution. Other survival functions can lead to the Fréchet distribution or negative Weibull distribution, with different constants $a_n,c_n$; the Gumbel distribution and the exponential example's normalization are not universal.

Independence is essential to the product calculation. If instead $X_1=X_2=\cdots =X$ for one rate-$\lambda$ exponential random variable, then $M_n=X$ for every $n$. Its distribution function stays $1-e^{-\lambda x}$ for $x\ge 0$ and zero below zero, and its median stays $(\log 2)/\lambda$. Perfect dependence creates no logarithmic growth: repeating the same random variable supplies no new chance of a larger value.

A historical starting point is R. A. Fisher and L. H. C. Tippett's 1928 paper on limiting laws of the largest or smallest sample member, published in volume 24, issue 2, pages 180–190 of the Proceedings of the Cambridge Philosophical Society. Its abstract studies the possible limiting forms and emphasizes that approach to a limit can be slow. ^[ref].

The standard Gumbel distribution for maxima has distribution function $\Lambda (z)=e^{-e^{-z}}$ for every real $z$. Its probability density function is the derivative of its distribution function:This article uses the maxima convention throughout. Reflecting a standard Gumbel distribution random variable $Z$ to $-Z$ gives the minima convention, with distribution function $1-\exp (-e^z)$. The two conventions have opposite directions of asymmetry, so their formulas must not be interchanged. ^[ref].

For a location $\mu \in \mathbb{R}$ and positive scale $b>0$, let $Y=\mu +bZ$. This location-scale family has cumulative distribution function and probability density functionSolving $\Lambda (z)=p$ gives the standard quantile function $q(p)=-\log (-\log p)$, $0<p<1$, and hence the standard medianThe median of $Y$ is therefore $\mu -b\log (\log 2)$. Location $\mu$ is not the median: at $x=\mu$ the cumulative distribution function is $e^{-1}\approx 0.367879$.

To derive the Gumbel distribution as an extreme-value limit, take the maximum of independent exponential random variables with rate $\lambda >0$ and setThe exact distribution function isThe finite-$n$ lower boundary is required because $M_n\ge 0$. For each fixed real $z$, eventually $z>-\log n$, and the natural logarithm of the first branch satisfiesExponentiating proves convergence in distribution to the standard Gumbel distribution. It is the familiar limit $(1-u/n{)}^n\to e^{-u}$ with $u=e^{-z}$: a simple calculation reveals a whole limiting shape. The name honors Emil Gumbel; Siegrist gives this exponential-maxima derivation and the standard Gumbel distribution formulas. ^[ref].

For the toy delay model $\lambda =1{\mathrm{s}}^{-1}$, $n=100$, the limiting Gumbel distribution approximation to $M_{100}$ has $\mu =(\log 100)\mathrm{s}$ and $b=1\mathrm{s}$. Its predicted median iscompared with the exact median $-\log (1-2^{-1/100})\mathrm{s}\approx 4.975147\mathrm{s}$. The approximation is about $0.003464$ seconds lower in this example. It is not an exact finite-sample probability distribution: the approximating Gumbel distribution assigns positive probability to negative delays, while the exact sample maximum cannot be negative.

The maximum domain of attraction of the Gumbel distribution is characterized by an exponential tail ratio on the scale of a Gumbel auxiliary function. This allows many original probability distributions to share the limiting shape, but it does not make the centering $\log n$ appropriate to all of them.