Relationship between the Poisson Process and the Gumbel Distribution

For a given sequence or collective, if:

  1. the interarrival times between events are well described by an exponential distribution
  2. the probability of ocurrence of events at any given year are described by a Poisson distribution
  3. and the duration of events is described by a Gamma distribution
then the sequence has been generated by a Poisson stochastic process

And, E. Gumbel showed that the distribution of maximum values taken from samples of an exponential distribution, converges to the Gumbel distribution. Therefore, the events' maximum interarrival times generated by a Poisson process, are Gumbel distributed.

References
Gumbel (1941)

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