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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to extreme value theory

Which statement about the Fisher–Tippett–Gnedenko theorem is correct?

The theorem states that if properly normalised maxima of iid variables converge to a non-degenerate distribution, that limit must be a GEV distribution. The type depends on the tail of the parent distribution, so it is not always Gumbel, and it concerns maxima rather than means.

  1. AIf suitably normalised block maxima of iid variables converge to a non-degenerate limit, that limit must be a GEV distribution.Correct
  2. BThe normalised sample mean of iid variables converges to a GEV distribution.
  3. CBlock maxima always converge to a Gumbel distribution whatever the parent distribution.
  4. DThe limit of normalised maxima is always a normal distribution when the variance is finite.
  5. The theorem applies only to parent distributions with finite upper endpoints.

Explanation

The theorem says that when normalised maxima have a non-degenerate limit, the limit belongs to the GEV family (Gumbel, Fréchet or Weibull type). It concerns maxima, not means, and the limit type depends on the parent tail. It is not restricted to bounded distributions.

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