The asymptotic distribution of magnitude trimmed sums for distributions in the Feller class |
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Authors: | Edward Whalen |
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Institution: | (1) Miles Pharmaceutical, S & DS, 400 Morgan Lane, 06516 West Haven, Connecticut |
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Abstract: | LetX, X
1,X
2,... be i.i.d. with common distribution functionF. Rather than study limit behavior of the sum,S
n
=X
1+ +X
n
, under constant normalizations, we consider the sum with ther
n
summands largest in magnitude removed from the sumS
n
, wherer
n
![rarr](/content/q33720672xl88655/xxlarge8594.gif) andr
n
n
–1 0. This is known as an intermediate magnitude trimmed sum. LetF be such that lim inft![rarr](/content/q33720672xl88655/xxlarge8594.gif) lim inf
t![rarr](/content/q33720672xl88655/xxlarge8594.gif)
EX
2
I(|X| t/)t
2
P((|X|>t)>0. The collection of suchF is known as the Feller class, a large class of distributions which includes all domains of attraction (in particular the stable laws). Pruitt(13) showed that asymptotic normality always holds for the trimmed sums ifF is in the Feller class and ifF is symmetric. Here, for anyF in the Feller class, we obtain complete results including the form of the possible limit laws and their convergence criteria, thus generalizing Pruitt's result to the asymmetric setting.This paper forms a portion of the author's Ph.D. dissertation under the supervision of Daniel C. Weiner. |
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Keywords: | Asymmetry Feller class nonnormal limits stochastic compactness trimmed sum |
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