Emil Julius Gumbel codified this theory in his 1958 book "Statistics of Extremes", including the Gumbel distributions that bear his name. These results can be extended to allowing for slight correlations between variables, but the classical theory does not extend to strong correlations of the order of the variance. One universality class of particular interest is that of log-correlated fields, where the correlations decay logarithmically with the distance. A summary of historically important publications relating to extreme value theory can be found in the article List of publications in statistics. Let formula_1 be a sequence of independent and identically distributed random variables with cumulative distribution function "F" and let formula_2 denote the maximum. In theory, the exact distribution of the maximum can be derived:  The associated indicator function formula_4 is a Bernoulli process with a success probability formula_5 that depends on the magnitude formula_6 of the extreme event. The number of extreme events within formula_7 trials thus follows a binomial distribution and the number of trials until an event occurs follows a geometric distribution with expected value and standard deviation of the same order formula_8. In practice, we might not have the distribution function formula_9 but the Fisher–Tippett–Gnedenko theorem provides an asymptotic result. If there exist sequences of constants formula_10 and formula_11 such that  as formula_13 then  where formula_15 depends on the tail shape of the distribution. When normalized, "G" belongs to one of the following non-degenerate distribution families:  Weibull law: formula_16 has a light tail with finite upper bound. Also known as Type 3. Gumbel law: formula_17 when the distribution of formula_18 has an exponential tail. Also known as Type 1  Fréchet Law: formula_19 when the distribution of formula_18 has a heavy tail (including polynomial decay). Also known as Type 2. In all cases, formula_21. Extreme value theory in more than one variable introduces additional issues that have to be addressed. One problem that arises is that one must specify what constitutes an extreme event. Although this is straightforward in the univariate case, there is no unambiguous way to do this in the multivariate case. The fundamental problem is that although it is possible to order a set of real-valued numbers, there is no natural way to order a set of vectors. As an example, in the univariate case, given a set of observations formula_22 it is straightforward to find the most extreme event simply by taking the maximum (or minimum) of the observations.