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 Electronic Communications in Probability > Vol. 12 (2007) > Paper 19 open journal systems 


Asymptotic results for empirical measures of weighted sums of independent random variables

Bernard Bercu, Universite Bordeaux 1
Wlodek Bryc, University of Cincinnati


Abstract
We investigate the asymptotic behavior of weighted sums of independent standardized random variables with uniformly bounded third moments. The sequence of weights is given by a family of rectangular matrices with uniformly small entries and approximately orthogonal rows. We prove that the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This result implies almost sure convergence of empirical periodograms, almost sure convergence of spectral distribution of circulant and reverse circulant matrices, and almost sure convergence of the CDF generated from independent random variables by independent random orthogonal matrices. In the special case of trigonometric weights, the speed of the almost sure convergence is described by a normal approximation as well as a large deviation principle.


Full text: PDF

Pages: 184-199

Published on: May 24, 2007
Bibliography
  1. Z. D. Bai, Methodologies in spectral analysis of large-dimensional random matrices, a review, Statist. Sinica 9 (1999), no. 3, 611-677, With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. MR1711663
  2. John R. Baxter and Naresh C. Jain, An approximation condition for large deviations and some applications, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 63-90. MR1412597
  3. I. Berkes and E. Csáki, A universal result in almost sure central limit theory, Stochastic Process. Appl. 94 (2001), 105-134. MR1835848
  4. Patrick Billingsley, Probability and measure, third ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1995, A Wiley-Interscience Publication. MR1324786 (95k:60001)
  5. Arup Bose, Sourav Chatterjee, and Sreela Gangopadhyay, Limiting spectral distributions of large dimensional random matrices, J. Indian Statist. Assoc. 41 (2003), no. 2, 221-259. MR2101995 (2005f:62027)
  6. Arup Bose and Joydip Mitra, Limiting spectral distribution of a special circulant, Statist. Probab. Lett. 60 (2002), no. 1, 111-120. MR1945684 (2003j:60043)
  7. G.A. Brosamler, An almost sure everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561-574. MR0957261 (89i:60045)
  8. Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, second ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. MR1619036 (99d:60030)
  9. Gilles Fay and Philippe Soulier, The periodogram of an i.i.d. sequence, Stochastic Process. Appl. 92 (2001), 315--343. MR1817591 (2001m:62100)
  10. Ion Grama and Michael Nussbaum, A functional Hungarian construction for sums of independent random variables, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 6, 923-957, En l'honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov. MR1955345 (2004c:60096)
  11. Fumio Hiai and Dénes Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000. MR1746976 (2001j:46099)
  12. Tiefeng Jiang, Maxima of entries of Haar distributed matrices, Probab. Theory Related Fields 131 (2005), no. 1, 121-144. MR2105046 (2005i:60013)
  13. Piotr Kokoszka and Thomas Mikosch, The periodogram at the Fourier frequencies, Stochastic Process. Appl. 86 (2000), no. 1, 49-79. MR1741196 (2000k:62174)
  14. M. T. Lacey and W. Phillip, A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), 201-205. MR1045184 (91e:60100)
  15. Russell Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J. 35 (1988), no. 3, 353-359. MR0978305 (90d:60038)
  16. Peter March and Timo Seppäläinen, Large deviations from the almost everywhere central limit theorem, J. Theoret. Probab. 10 (1997), no. 4, 935-965. MR1481655 (98m:60040)
  17. Adam Massey, Steven J. Miller, and John Sinsheimer, Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices, Journ. Theoret. Probab. (2005), (to appear), arXiv:math.PR/0512146.
  18. Luca Pratelli Patrizia Berti and Pietro Rigo, Almost sure weak convergence of random probability measures, Stochastics: An International Journal of Probability and Stochastic Processes 78 (2006), 91-97. MR2236634
  19. A. I. Sakhanenko, Rate of convergence in the invariance principle for variables with exponential moments that are not identically distributed, Limit theorems for sums of random variables, Trudy Inst. Mat. (English translation in: Limit theorems for sums of random variables, eds. Balakrishna and A.A.Borovkov. Optimization Software, New York, 1985.), vol. 3, ``Nauka'' Sibirsk. Otdel., Novosibirsk, 1984, pp. 4-49. MR0749757 (86g:60047)
  20. A. I. Sakhanenko, On the accuracy of normal approximation in the invariance principle [translation of Trudy Inst. Mat. (Novosibirsk) 13 (1989), Asimptot. Analiz Raspred. Sluch. Protsess., 40-66; MR 91d:60082], Siberian Adv. Math. 1 (1991), no. 4, 58-91, Siberian Advances in Mathematics. MR1138005
  21. P. Schatte, On strong versions of the almost sure central limit theorem, Math. Nachr. 137 (1988), 249-256. MR0968997 (89i:60070)
















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Electronic Communications in Probability. ISSN: 1083-589X