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A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. (English) Zbl 0657.60029
Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables with common non-degenerate distribution function F. For each n let \(X_{1,n}\leq...\leq X_{n,n}\) denote the order statistics based on \(X_ 1,...,X_ n\). For fixed non-negative integers m and k the lightly trimmed sum is defined by \(S_ n(m,k)=\sum^{n-k}_{j=m+1}X_{j,n}\). When \(m=k=0\) this becomes an ordinary untrimmed sum.
The authors present a unified probabilistic approach to the problem of asymptotic distribution of lightly trimmed (as a special case, untrimmed) sums. This approach is based upon the asymptotic behavior of the uniform empirical distribution function in conjunction with an integral representation of these sums.
At first they prove general theorems giving necessary and sufficient conditions for the convergence in law of normalized trimmed (and untrimmed) sums along subsequences of positive integers. Some of these results also show which portions of these sums contribute to ingredients of the limiting laws.
The authors also give a representation of the infinitely divisible random variable in terms of Poisson processes, which is an integral part of their approach. In a sequence of corollaries to these general theorems the authors give necessary and sufficient conditions for these lightly trimmed or untrimmed sums to be in the domain of attraction or partial attraction of a normal or non-normal stable law, in the domain of partial attraction of some infinitely divisible law and a necessary and sufficient condition for the stochastic compactness and subsequential compactness of these sums.
The analytic conditions in these theorems and corollaries are all expressed in terms of quantile function, i.e. left continuous inverse of F.
Reviewer: T.Mori

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
Full Text: DOI
[1] Bingham, N.H; Goldie, C.M; Teugels, J.L, Regular variation, (1987), Cambridge Univ. Press Cambridge · Zbl 0617.26001
[2] Csörgö, M; Csörgö, S; Horváth, L; Mason, D.M, Weighted empirical and quantile processes, Ann. probab., 14, 31-85, (1986) · Zbl 0589.60029
[3] Csörgö, M; Csörgö, S; Horváth, L; Mason, D.M, Normal and stable convergence of integral functions of the empirical distribution function, Ann. probab., 14, 86-118, (1986) · Zbl 0589.60030
[4] \scS. Csörgö, E. Haeusler and D.M. Mason, The asymptotic distribution of trimmed sums, Ann. Probab., to appear.
[5] Csörgö, S; Horváth, L; Mason, D.M, What portion of the sample makes a partial sum asymptotically stable or normal?, Probab. theory relat. fields, 72, 1-16, (1986) · Zbl 0572.60028
[6] 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
[7] deHaan, L; Resnick, S.I, Stochastic compactness and point processes, J. austral. math. soc. ser. A, 37, 307-316, (1984) · Zbl 0555.60018
[8] Doeblin, W, Sur l’ensemble de puissances d’une loi de probabilité, Studio math., 9, 71-96, (1940) · Zbl 0063.01128
[9] Feller, W, Über den zentralen grenzwertsatz der wahrscheinlichkeitsrechnung, Math. Z., 40, 521-559, (1935) · JFM 61.0558.03
[10] Feller, W, ()
[11] Feller, W, On regular variation and local limit theorems, (), 373-388, Part 1 · Zbl 0214.17303
[12] Gnedenko, B.V, On the theory of domains of attractions of stable laws, Uchen. zap. moskov. GoS. univ. mat., 30, 61-81, (1939), [Russian]
[13] Gnedenko, B.V, Some theorems on the powers of distribution functions, Uchen. zap. moskov. GoS. univ. mat., 45, 61-72, (1940), [Russian]
[14] Gnedenko, B.V; Kolmogorov, A.N, Limit distributions for sums of independent random variables, (1954), Addison-Wesley Reading, MA · Zbl 0056.36001
[15] Goldie, C.M; Seneta, E, On domains of partial attraction, J. austral. math. soc. ser. A, 32, 328-331, (1982) · Zbl 0499.60021
[16] Jain, N.C; Orey, S, Domains of partial attraction and tightness conditions, Ann. probab., 8, 584-599, (1980) · Zbl 0442.60050
[17] Khinchin, A.Ya, Sul dominio di attrazione Della legge di Gauss, Giorn. ist. ital. attuary, 6, 371-393, (1935) · Zbl 0013.02901
[18] Khinchin, A.Ya, Zur theorie der unbeschränkt teilbaren verteilungsgesetze, Mat. sb. N.S., 2, 44, 79-120, (1937) · JFM 63.0498.02
[19] \scJ. Kuti, personal communication, 1985.
[20] LePage, R; Woodroofe, M; Zinn, J, Convergence to a stable distribution via order statistics, Ann. probab., 9, 624-632, (1981) · Zbl 0465.60031
[21] Lévy, P, Calcul des probabilités, (1925), Gauthier-Villars Paris · JFM 51.0380.02
[22] Lévy, P, Propriétés asymptotiques des sommes de variables aléatoires indépendantes on enchaînées, J. math. pures appl. (9), 14, 347-402, (1935) · JFM 61.1291.01
[23] Lévy, P, Théorie de l’addition des variables aléatoires, (1937), Gauthier-Villars Paris · JFM 63.0490.04
[24] Loève, M, Probability theory, (1955), Van Nostrand Toronto · Zbl 0108.14202
[25] Loève, M, Ranking limit problem, (), 177-194
[26] Maller, R.A, A note on domains of partial attraction, Ann. probab., 8, 576-583, (1980) · Zbl 0447.60019
[27] Maller, R.A, Some properties of stochastic compactness, J. austral. math. soc. ser. A, 30, 264-277, (1981) · Zbl 0468.60028
[28] Maller, R.A, Some results on asymptotics of trimmed means in projection pursuit preprint, (1986)
[29] Mason, D.M, The asymptotic distribution of generalized Rényi statistics, Acta. sci. math. (Szeged), 48, 315-323, (1986) · Zbl 0592.62016
[30] Mori, T, On the limit distributions of lightly trimmed sums, (), 507-516 · Zbl 0552.60018
[31] Pruitt, W.E, The class of limit laws for stochastically compact normed sums, Ann. probab., 11, 962-969, (1983) · Zbl 0519.60014
[32] Root, D; Rubin, H, A probabilistic proof of the normal convergence criterion, Ann. probab., 1, 867-869, (1973) · Zbl 0271.60030
[33] Rossberg, H.-J, Über das asymptotische verhalten der rand- und zentralglieder einer variationsreihe (II), Publ. math. debrecen, 14, 83-90, (1967) · Zbl 0183.21501
[34] Sato, K, A note on-infinitely divisible distributions and their Lévy measures, Sci. rep. Tokyo kyoiku daigaku sect. A, 12, 101-109, (1973) · Zbl 0279.60010
[35] Seneta, E, Sequential criteria for regular variation, Quart. J. math. Oxford 2, 22, 565-570, (1971) · Zbl 0229.26004
[36] Seneta, E, Regularly varying functions, () · Zbl 0291.60043
[37] Simons, G; Stout, W, A weak invariance principle with applications to domains of attraction, Ann. probab., 6, 294-315, (1978) · Zbl 0376.60027
[38] Tucker, H.G, On the asymptotic independence of the partial sums of positive and negative parts of independent random variables, Adv. appl. probab., 3, 404-425, (1971) · Zbl 0222.60033
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