The asymptotic distribution of randomly weighted sums and self-normalized sums

Peter Kevei (University of Szeged)
David M Mason (University of Delaware)


We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where $\{ Y_{i} : i\geq 1 \}$ are non-negative i.i.d.~random variables, and $\{ X_{i} : i\geq 1 \}  $ are i.i.d. random variables, independent of $\{ Y_{i} : i \geq 1 \}$. The main result of the paper is that each subsequential limit law of $T_n$ is continuous for any non-degenerate $X_1$ with finite expectation, if and only if $Y_1$ is in the centered Feller class.

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Pages: 1-21

Publication Date: June 18, 2012

DOI: 10.1214/EJP.v17-2092


  • Breiman, L. On some limit theorems similar to the arc-sin law. Teor. Verojatnost. i Primenen. 10 1965 351--360. MR0184274
  • Breiman, Leo. Probability. Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont. 1968 ix+421 pp. MR0229267
  • Cline, Daren B. H. Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72 (1986), no. 4, 529--557. MR0847385
  • Darling, D. A. The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73, (1952). 95--107. MR0048726
  • de Haan, L. On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts, 32 Mathematisch Centrum, Amsterdam 1970 v+124 pp. (loose errata). MR0286156
  • Denisov, Denis; Zwart, Bert. On a theorem of Breiman and a class of random difference equations. J. Appl. Probab. 44 (2007), no. 4, 1031--1046. MR2382943
  • Giné, Evarist; Götze, Friedrich; Mason, David M. When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 (1997), no. 3, 1514--1531. MR1457629
  • Griffin, Philip S. Matrix normalized sums of independent identically distributed random vectors. Ann. Probab. 14 (1986), no. 1, 224--246. MR0815967
  • Griffin, Philip S. Tightness of the Student $t$-statistic. Electron. Comm. Probab. 7 (2002), 181--190 (electronic). MR1937903
  • Jessen, Anders Hedegaard; Mikosch, Thomas. Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80(94) (2006), 171--192. MR2281913
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2 MR1876169
  • Maller, R. A. Relative stability, characteristic functions and stochastic compactness. J. Austral. Math. Soc. Ser. A 28 (1979), no. 4, 499--509. MR0562881
  • Maller, Ross; Mason, David M. Convergence in distribution of Lévy processes at small times with self-normalization. Acta Sci. Math. (Szeged) 74 (2008), no. 1-2, 315--347. MR2431109
  • Maller, Ross; Mason, David M. Stochastic compactness of Lévy processes. High dimensional probability V: the Luminy volume, 239--257, Inst. Math. Stat. Collect., 5, Inst. Math. Statist., Beachwood, OH, 2009. MR2797951
  • Maller, Ross; Mason, David M. Small-time compactness and convergence behavior of deterministically and self-normalised Lévy processes. Trans. Amer. Math. Soc. 362 (2010), no. 4, 2205--2248. MR2574893
  • Mason, David M. The asymptotic distribution of self-normalized triangular arrays. J. Theoret. Probab. 18 (2005), no. 4, 853--870. MR2289935
  • Mason, David M.; Newton, Michael A. A rank statistics approach to the consistency of a general bootstrap. Ann. Statist. 20 (1992), no. 3, 1611--1624. MR1186268
  • Mason, David M.; Zinn, Joel. When does a randomly weighted self-normalized sum converge in distribution? Electron. Comm. Probab. 10 (2005), 70--81 (electronic). MR2133894
  • Meerschaert, Mark M.; Scheffler, Hans-Peter. Limit distributions for sums of independent random vectors. Heavy tails in theory and practice. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, 2001. xvi+484 pp. ISBN: 0-471-35629-8 MR1840531

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