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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].

A non-degenerate distribution function is max-stable when every sample maximum of $n$ independent and identically distributed random variables can be positively scaled and centered to have the original law. Thus for every integer $n\ge 1$ there are $a_n>0,b_n\in \mathbb{R}$ with $F(a_{n}x+b_n{)}^n=F(x)$. The extremal types theorem classifies these laws, up to affine changes, as the Gumbel distribution, Fréchet distribution, and negative Weibull distribution.

A nondegenerate distribution function $G$ is a max-stable distribution if, for every integer $r\ge 1$, there are $a_r>0$ and $b_r\in \mathbb{R}$ such thatEquivalently, the suitably centered and positively scaled sample maximum of $r$ independent and identically distributed random variables with law $G$ again has law $G$. The extremal types theorem states that every nondegenerate limiting law of normalized sample maxima is a max-stable distribution. Thus this property identifies all possible limiting types, although a particular starting distribution function need not admit any such limiting normalization.

The three standard types are the Gumbel distributionthe Fréchet distribution, with $\alpha >0$,and the negative Weibull distribution, with $\alpha >0$,They are max-stable distributions, as follows respectively by taking $(a_r,b_r)=(1,\log r)$, $(r^{1/\alpha },0)$, and $(r^{-1/\alpha },0)$. The necessary and sufficient classification is: $G$ is max-stable exactly when $G(x)=H((x-b)/a)$ for $a>0$, $b\in \mathbb{R}$, and one of these three types $H$. A positive affine map sends a max-stable distribution to another max-stable distribution.

For the requested threshold equivalence, set $p_n=1-F(u_n)$. By independence,If $n{p}_n\to \tau <\infty$, then $p_n\to 0$ and the expansion $\log (1-p)=-p+O(p^2)$ gives $n\log (1-p_n)\to -\tau$, since $n{p}_n^2=(n{p}_n)p_n\to 0$. Exponentiation proves the claimed limit. Conversely, if $(1-p_n{)}^n\to e^{-\tau }>0$, then $p_n\to 0$: any subsequence with $p_n\ge \delta >0$ would have its powers tending to zero. Taking logarithms now gives $-n\log (1-p_n)\to \tau$. Since $-\log (1-p)/p\to 1$, including the continuous extension at $p=0$, it follows that $n{p}_n\to \tau$. Hence

The exceedance count $S_n$ has the binomial distribution with parameters $n,p_n$. For $\tau >0$ and each fixed nonnegative integer $s$, its probability mass function satisfieswhere $(n{)}_s=n(n-1)\cdots (n-s+1)$ and $(n{)}_0=1$. Indeed $(n{)}_s/n^s\to 1$, $(n{p}_n{)}^s\to {\tau }^s$, and $(1-p_n{)}^{n-s}\to e^{-\tau }$. If $\tau =0$, the bound $\mathbb{P}(S_n\ge 1)\le \mathbb{E}{S}_n=n{p}_n\to 0$ gives the degenerate Poisson distribution at zero. Summing the finitely many terms proves, in both cases,This derives the needed instance of the Poisson limit theorem directly.

Finally fix $x$ with $G(x)>0$ and put $u_n=a_{n}x+b_n$. The assumed limit for the normalized sample maximum and the proved equivalence give $n{p}_n\to -\log G(x)$. The second-largest order statistic is at most $u_n$ exactly when at most one observation is strictly greater than $u_n$. This event identity holds even if observations have ties. Apply the preceding Poisson limit theorem with $k=1$ to obtainThe endpoint $G(x)=1$ corresponds to $\tau =0$ and is already included. This is the case $r=2$ of the exceedance count limit for extreme order statistics.

A non-degenerate probability distribution is not concentrated at a single point. In extreme value theory, a non-degenerate distribution function $F$ is max-stable if, for each integer $n\ge 1$, constants $a_n>0,b_n\in \mathbb{R}$ satisfy $F(a_{n}x+b_n{)}^n=F(x)$ for every $x$. This says that a normalized sample maximum has the same law as one observation. A distribution function $F$ belongs to the maximum domain of attraction of a non-degenerate $G$ if there are $a_n>0,b_n$ such thatat every continuity point of $G$. Equivalently, $(M_n-b_n)/a_n$ has convergence in distribution to $G$.

The extremal types theorem says that any such non-degenerate limit is max-stable and, up to a positive affine change of variable, is exactly one of the following:These are respectively the Gumbel distribution, Fréchet distribution, and negative Weibull distribution. The theorem is also known as the Fisher–Tippett–Gnedenko theorem.

Useful sufficient conditions can be expressed using the survival function $\overline{F}=1-F$ and the right endpoint of a distribution $x_F$. An infinite endpoint with $\overline{F}(tx)/\overline{F}(t)\to x^{-\alpha }$ for every $x>0$ gives the Fréchet distribution domain. A finite endpoint with $\overline{F}(x_F-sx)/\overline{F}(x_F-s)\to x^{\alpha }$ as $s\downarrow 0$ gives the negative Weibull distribution domain. For the Gumbel distribution, it suffices that a positive Gumbel auxiliary function $a(t)$ gives $\overline{F}(t+a(t)x)/\overline{F}(t)\to e^{-x}$ for every real $x$ as $t\uparrow x_F$. These are regular variation and exponential tail-ratio conditions; they need no proof here.

In case (i), ${\overline{F}}_1(x)=(1-x{)}^2$ on $0<x<1$, so the finite-endpoint ratio is exactly $x^2$. Therefore the domain is negative Weibull with shape $2$. With $a_n=n^{-1/2}$ and $b_n=1$,and the limit is one for $x\ge 0$.

In case (ii), ${\overline{F}}_2(t)=e^{-\lambda t}$ has exponential tail ratio with the constant Gumbel auxiliary function $1/\lambda$. Therefore the domain is Gumbel. Choosing $a_n=1/\lambda$ and $b_n=\log n/\lambda$ givesIn case (iii), ${\overline{F}}_3(t)=(1+t{)}^{-\lambda }$ has regular variation of index $-\lambda$, so the domain is Fréchet with shape $\lambda$. The convenient choices $a_n=n^{1/\lambda }$ and $b_n=-1$ givewith limit zero for $x\le 0$.

Finally, define the empirical distribution function $F_n(x)=n^{-1}{\sum }_{i=1}^n{1}_{\{X_i\le x\}}$. On a sample with positive threshold $t=X_{(n-k)}$ and exactly $k$ observations strictly above $t$, its survival function has $1-F_n(t)=k/n$. Using the Tonelli theorem to integrate the finite nonnegative sum givesThis is the Hill estimator as an empirical plug-in version of the tail-integral limit. It does not prove consistency for fixed $k$; that is a separate issue.

There is a finite-sample qualification because the question permits ties and does not assume positive observations. The formula requires $1\le k<n$ and $t>0$. If $r=\#\{i:X_i>t\}<k$ because the threshold is tied, direct substitution giveswhere terms equal to the threshold contribute zero. This is generally $k/r$ times the displayed Hill estimator, rather than the same estimator. If $r=0$, the empirical plug-in denominator is zero. For a continuous probability distribution the no-tie condition holds almost surely; for a Fréchet distribution domain and fixed $k$, the threshold is positive with probability tending to one. Thus the stated plug-in identity holds at a positive untied threshold, and the literal claim for all samples from an arbitrary distribution function needs this qualification.