Hsu-Robbins and Spitzer's theorems for the variations of fractional Brownian motion

Ciprian A. Tudor (Universte de Paris 1)

Abstract


Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.

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Pages: 278-289

Publication Date: July 9, 2009

DOI: 10.1214/ECP.v14-1481

References

  1. P.G. Doyle, J.L. Snell. Random walks and electric networks. Carus Mathematical Monographs 22 (1984) Math. Assoc. America. Math. Review 89a:94023
  2. J.M. Albin. A note on Rosenblatt distributions. Statistics and Probability Letters 40(1) (1998), 83-91. Math. Review MR1650532 (2000b:60101)
  3. J.-C. Breton and I. Nourdin. Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. Electronic Communications in Probability 13 (2008), 482-493. Math. Review MR2447835
  4. P. Erd\"os. On a theorem of Hsu and Robbins. Ann. Math. Statistics 20 (1949), 286-291. Math. Review MR0030714 (11,40f)
  5. P. Erd\"os. Remark on my paper "On a theorem of Hsu and Robbins." Ann. Math. Statistics 21 (1950), 138. Math. Review MR0032970 (11,375b)
  6. C.C. Heyde. A supplement to the strong law of large numbers Journal of Applied Probability 12 (1975), 173-175. Math. Review MR0368116 (51 #4358)
  7. H-C. Ho and T. Hsing. Limit theorems for functionals of moving averages. The Annals of Probability 25(4) (1997), 1636-1669. Math. Review MR1487431 (98m:60027)
  8. P. Hsu and H. Robbins. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31. Math. Review MR0019852 (8,470e)
  9. P. Major. Tail behaviour of multiple integrals and U-statistics. Probability Surveys. 2 (2005), 448-505. Math. Review MR2203678 (2007g:60036)
  10. I. Nourdin, D. Nualart and C.A. Tudor. Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Preprint, 2008. Math. Review number not available.
  11. I. Nourdin and G. Peccati. Stein's method on Wiener chaos. Preprint, to appear in Probability Theory and Related Fields, (2009). Math. Review number not available.
  12. D. Nualart. Malliavin calculus and Related Topics. Springer, (2006). Math. Review MR2200233 (2006j:60004)
  13. A. Spataru. Precise asymptotics in Spitzer's law of large numbers. Journal of Theoretical Probability 12(3) (1999), 811-819. Math. Review MR1702879 (2000j:60035)
  14. F. Spitzer. A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc., 82 (1956), 323-339. Math. Review MR0079851 (18,156e)
  15. W.B. Wu. Unit root testing for functionals of linear processes. Econometric Theory, 22(1) (2006), 1-14. Math. Review MR2212691 (2006k:62066)
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