What's the Distribution of the Maximum of IID Variables When m is Large?

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Discussion Overview

The discussion revolves around the distribution of the maximum of m independent and identically distributed (IID) random variables as m becomes large. Participants explore the behavior of the cumulative distribution function (CDF) of the maximum variable and its convergence properties under different conditions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes that the CDF of the maximum Y, denoted as P(Y
  • Another participant questions whether the CDF of Y converges to 0, suggesting that this depends on the distribution of the Yk variables.
  • It is noted that if the Yk are distributed such that P(Yk < y) < 1, the CDF may converge to 0, but if the Yk are bounded, this is not the case.
  • A participant highlights that the maximum of a sample from an exponential distribution will converge to the Gumbel distribution when appropriately standardized.
  • Reference is made to a statistical text that discusses extreme value statistics, indicating that the result is contingent on the underlying distribution of the Y variables.

Areas of Agreement / Disagreement

Participants express differing views on the convergence behavior of the CDF of the maximum variable, indicating that the discussion remains unresolved with multiple competing perspectives on the conditions affecting convergence.

Contextual Notes

Limitations include the dependence on the specific distributions of the Yk variables and the conditions under which convergence is assessed. The discussion does not resolve the mathematical steps involved in determining the distribution of the maximum.

Pascal22
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Suppose that there are m independent and identically distributed variables Y1, Y2, ... Ym. Yi - are random variables. Let Y denote the maxof Y1, Y2, ... Ym. What's the distribution of Y when m is very big?

Thank you for any help, in advance.
 
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P(Y<y) = P(Y1<y)P(Y2<y)...P(Ym<y) = P(Y1<y)m (independent and identially distributed).
 
Thanks a lot!

Do I understand correctly? That CDF of Y will converge 0?
 
Last edited:
Pascal22 said:
Thanks a lot!

Do I understand correctly? That CDF of Y will converge 0?

It depends. If the Yk are distributed so that P(Yk < y) < 1, yes. However if the Yk are bounded, then no.
 
"It depends. If the Yk are distributed so that P(Yk < y) < 1, yes"

I don't think I'm getting your point, or perhaps I'm looking a little to picky-like. If you are simply looking at the value of the probability, then the comment makes sense. But remember, for example, that the appropriately standardized distribution for the max of an SRS from an exponential will converge to the Gumbel distribution.
 
Have a look at D.R. Cox and D.V. Hinkley, Theoretical Statistics, chapter A2.5 "Extreme value statistics". The result depends on the underlying statistics of the Y.
 

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