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

[Pascal22]

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.

 

[mathman]

P(Y<y) = P(Y1<y)P(Y2<y)...P(Ym<y) = P(Y1<y)^m (independent and identially distributed).

 

[Pascal22]

Thanks a lot!

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

 

[mathman]

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.

 

[statdad]

"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.

 

[DrDu]

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.