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Part BCSIR NET December 2023extreme-value-scaling

Extreme value scaling

Let be i.i.d. with CDF F(x) = 0 for x < 5 and 1 − for x ≥ 5. Define {} and , and let Z be standard normal. Which one of the following statements is true?

  1. A.
  2. B. in probability as ✓
  3. C. in distribution as
  4. D.

Solution

~ Exp(n), so in probability while in probability (the right scaling is n, not .

The trap it tests

Moments and tails

A moment assumed to exist, or tail behaviour assumed to be tame.

Drill statements like this

Related counterexample: Convergence in probability implies almost sure convergence

More on this topic

From Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

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