This note is of a didactic nature. We will derive the limit distributions for maxima of i.i.d. random variables and give sufficient conditions for their domains of attraction.  These results can be found in the works of GNEDENKO and VON MISES. The present exposition contains new proofs which are intended to be sufficiently simple to be used in an elementary course of probability theory. 2 The limit distributions We will be concerned with the following problem. Suppose X I , X,, ... are independent real-valued random variables with common distribution function (df) F . We define for n = 1, 2, .. Y, = max(X,, X2, ...,X,). We remark that one can interpret all the results that follow as results for minima by noting that min (XI, ..., X,)= --ax ( - X I , ..., -X,,). It follows from the independence of the X i that P(Y, < x } = P ( X , < x , ..., x, < x } = F ( x ) . a,,> 0 and b,(n = 1,2, ...) such that the sequence P(u,- '( Y, We ask for conditions which enables one to choose sequences of real constants
Statistica Neerlandica – Wiley
Published: Dec 1, 1976
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