Regular stability of large order statistics

✍ Scribed by R.J. Tomkins

• Book ID: 104302917
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 335 KB
• Volume: 41
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(98)00132-1

✦ Synopsis

For r >/1, let Mnr be the rth largest of {X1,... ,X,}, where XI,X2 .... are i.i.d, random variables with common distribution function F such that F(x)< 1 for all x. Define #(x) =F 1(1 -x -~), x>~ 1. {M,r,n>~r} is called regularly stable if Mnr/12(n)--~ 1 in probability and (*)#(n t)/#(n) ~ h(t)< ve for every t > 1. Several sets of necessary and sufficient conditions for regular stability are presented. In particular, {Mnr} is regularly stable if and only if (.) holds, and regular stability with h(t)-= 1 is tantamount to complete stability. Moreover, (.) always implies the almost sure stability of {M,r},r>~ 1.