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Distribution and moment convergence of martingales
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  • Published: September 1988

Distribution and moment convergence of martingales

  • H. Teicher1 

Probability Theory and Related Fields volume 79, pages 303–316 (1988)Cite this article

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Summary

If \(S_{n,k} = \mathop \Sigma \limits_{1 \leqq i_1 < i_k \leqq m_n } X_{ni_1 } ...{\text{ }}X_{ni_k } \) where {X n j ,ℱ n j 1≦j≦m n ↑∞, n≧1} is a martingale difference array, conditions are given for the distribution and moment convergence of S n,k to the distribution and moments of \(\frac{1}{{k!}}H_k (Z)\) where H k is the Hermite polynomial of degree k and Z is a standard normal variable. This is intimately related to an identity (*) for multiple Wiener integrals. Under alternative conditions, similar results hold for S n, k /U k n and S n, k /V k n where \(U_n^2 = \sum\limits_{j = 1}^{m_n } {X_{n j}^2 }\) and V 2 n V 2 n is the conditional variance.

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References

  1. Aldous, D.J., Eagleson, G.K.: On mixing and stability of limit theorems. Ann. Probab. 6, 325–331 (1978)

    MathSciNet  Google Scholar 

  2. Alvo, M., Cabilio, P., Feigin, P.: A class of martingales with non-symmetric limit distributions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 58, 87–93 (1981)

    Article  MathSciNet  Google Scholar 

  3. Bernstein, S.: Quelques remarques sur le théoreme limite Liapounoff. Dokl. Akad. Nauk SSSR 24, 3–8 (1939)

    MATH  Google Scholar 

  4. Brown, B.: Characteristic functions, moments and the central limit theorem. Ann. Math. Stat. 41, 658–664 (1970)

    MATH  Google Scholar 

  5. Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab. 1, 19–42 (1973)

    MATH  MathSciNet  Google Scholar 

  6. Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)

    MathSciNet  Google Scholar 

  7. Dvoretzky, A.: Asymptotic normality for sums of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Prob., pp. 513–535 Berkeley: University of California Press (1972)

    Google Scholar 

  8. Engel, D.D.: The multiple stochastic integral. Mem. Am. Math. Soc. 38 (1982)

  9. Gänssler, P., Häusler, E.: Remarks on the functional central limit theorem for martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 237–243 (1979)

    Article  Google Scholar 

  10. Hall, P.: The convergence of moments in the martingale central limit theorem. Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, 253–260

  11. Hall, P., Heyde, C.C.: Martingale limit theory and its application. New York: Academic Press 1980

    Google Scholar 

  12. Helland, I.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat. 9, 79–94 (1982)

    MATH  MathSciNet  Google Scholar 

  13. Ito, K. Multiple Wiener integral. J. Math. Soc. Japan 3, 157–169 (1951)

    MATH  MathSciNet  Google Scholar 

  14. Major, P.: Multiple Wiener-Ito integrals (Lect. Notes Math vol. 849. Berlin Heidelberg New York: 1981

  15. McLeish, D.L.: Dependent central limit theorems and invariance principles. Ann. Probab. 2, 620–628 (1974)

    MATH  MathSciNet  Google Scholar 

  16. Raikov, D.A.: On a connection between the central limit theorem in the theory of probability and the law of large numbers. Izvestiya Akad. Nauk SSSR, Ser. Mat. pp. 323–338 (1938)

  17. Rootzen, H.: On the functional central limit theorem for martingales II. Z. Wahrscheinlichkeitstheor. Verw. Geb. 51, 79–93 (1980)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Hill Center for Mathematical Sciences, Rutgers University, 08903, New Brunswick, NJ, USA

    H. Teicher

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  1. H. Teicher
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Research supported by the National Science Foundation under Grant DMS-8601346

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Teicher, H. Distribution and moment convergence of martingales. Probab. Th. Rel. Fields 79, 303–316 (1988). https://doi.org/10.1007/BF00320924

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  • Received: 01 March 1987

  • Revised: 10 March 1988

  • Issue Date: September 1988

  • DOI: https://doi.org/10.1007/BF00320924

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Normal Variable
  • Conditional Variance
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