×

zbMATH — the first resource for mathematics

What portion of the sample makes a partial sum asymptotically stable or normal? (English) Zbl 0572.60028
Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index \(0<\alpha \leq 2\) be given. We show that if at each stage n a number \(k_ n\) depending on n of the lower and upper order statistics are removed from the \(n^{th}\) partial sum of the given random variables then under appropriate conditions on \(k_ n\) the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.

MSC:
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arov, D.Z., Bobrov, A.A.: The extreme terms of a sample and their role in the sum of independent variables. Theor. Probability Appl. 5, 377–396 (1960) · Zbl 0098.11202 · doi:10.1137/1105038
[2] Balkema, A., De Haan, L.: Limit laws for order statistics. In: Colloquia Math. Soc. J. Bolyai 11. Limit Theorems of Probability (P. Révész, Ed.), pp. 17–22. Amsterdam: North-Holland 1975
[3] Csörgo, M., Csörgo, S., Horváth, L., Masson, D.M.: Weighted empirical and quantile processes. Ann. Probability. To appear
[4] Csörgo, M., Csörgo, S., Horváth, L., Mason, D.M.: Normal and stable convergence of integral functionals of the empirical distribution fuctions. Ann. Probability. To appear
[5] Darling, D.A.: The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73, 95–107 (1952) · Zbl 0047.37502 · doi:10.1090/S0002-9947-1952-0048726-0
[6] De Haan, L.: On regular variation and its application to the weak convergence of sample extremes. Amsterdam: Mathematical Centre Tracts 32, 1970 · Zbl 0226.60039
[7] Hall, P.: On the extreme terms of a sample from the domain of attraction of a stable law. J. London Math. Soc. 18, 181–191 (1978) · Zbl 0387.60029 · doi:10.1112/jlms/s2-18.1.181
[8] Maller, R.A.: Asymptotic normality of lightly trimmed means – a converse. Math. Proc. Cambridge Philos. Soc. 92, 535–545 (1982) · Zbl 0534.60030 · doi:10.1017/S0305004100060229
[9] Rossberg, H.J.: Über das asymptotische Verhalten der Rand- und Zentralglieder einer Variationsreihe (II). Publ. Math. Debrecen 14, 83–90 (1967) · Zbl 0183.21501
[10] Stigler, S.M.: The asymptotic distribution of the trimmed mean. Ann. Statist. 1, 472–477 (1973) · Zbl 0261.62016 · doi:10.1214/aos/1176342412
[11] Teugels, J.L.: Limit theorems on order statistics. Ann. Probability 9, 868–880 (1981) · Zbl 0467.62046 · doi:10.1214/aop/1176994314
[12] Tucker, H.G.: On the asymptotic independence of the partial sums of positive and negative parts of independent random variables. Advances in Appl. Probability 3, 404–425 (1971) · Zbl 0222.60033 · doi:10.2307/1426178
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.