Editing Gumbel distribution (section)

===The discrete Gumbel distribution===
Many problems in [[discrete mathematics]] involve the study of an extremal parameter that follows a discrete version of the Gumbel distribution.<ref name=AguechAlthagafiBanderier>
{{Citation |arxiv=2311.13124|title=Height of walks with resets, the Moran model, and the discrete Gumbel distribution  |year=2023|first1=R.|last1=Aguech|first2=A.|last2=Althagafi|first3=C.|last3=Banderier|journal=Séminaire Lotharingien de Combinatoire|volume=87B|issue=12|pages=1–37}}</ref><ref>''Analytic Combinatorics'', Flajolet and Sedgewick.</ref> This ''discrete'' version is the law of <math>Y = \lceil X \rceil</math>, where <math>X</math> follows the ''continuous'' Gumbel distribution <math>\mathrm{Gumbel}(\mu, \beta)</math>.
Accordingly, this gives <math>P(Y \leq h) = \exp(-\exp(-(h-\mu)/\beta))</math> for any <math>h \in \mathbb Z</math>.

Denoting <math>\mathrm{DGumbel}(\mu, \beta)</math> as the discrete version, one has <math>\lceil X \rceil \sim \mathrm{DGumbel}(\mu, \beta)</math> and <math>\lfloor X \rfloor \sim \mathrm{DGumbel}(\mu - 1, \beta)</math>.

There is no known closed form for the mean, variance (or higher-order moments) of the discrete Gumbel distribution, but it is easy to obtain high-precision numerical evaluations via rapidly converging infinite sums. For example, this yields <math>{\mathbb E}[\mathrm{DGumbel}(0,1)]=1.077240905953631072609...</math>, but it remains an open problem to find a closed form for this constant (it is plausible there is none).

Aguech, Althagafi, and Banderier<ref name=AguechAlthagafiBanderier/> provide various bounds linking the discrete and continuous versions of the Gumbel distribution and explicitly detail (using methods from [[Mellin transform]]) the oscillating phenomena that appear when one has a sequence of random variables <math>\lfloor Y_n - c \ln n \rfloor</math> converging to a discrete Gumbel distribution.