The central limit theorem for summability methods of I.I.D. random variables

  • Paul Embrechts
  • Makoto Maejima
Article

Keywords

Stochastic Process Probability Theory Limit Theorem Mathematical Biology Central Limit 

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References

  1. 1.
    Azlarov, T.A., Meredov, B.: Some estimates in the limit theorem for the Abel summability of random variables. (In Russian.) Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1977, nℴ 5, 7–15 (1977)Google Scholar
  2. 2.
    Bikelis, A.: Estimates of the remainder term in the central limit theorem. (In Russian.) Litovsk. Mat. Sb. 6, 323–346 (1966)Google Scholar
  3. 3.
    Bikelis, A.: On the accuracy of Gaussian approximation to the distribution of sums of independent identically distributed random variables. (In Russian.) Litovsk. Mat. Sb. 11, 237–240 (1971)Google Scholar
  4. 4.
    Bikelis, A.: Asymptotic expansions of sums of independent m-lattice random vectors. (In Russian.) Litovsk. Mat. Sb. 12, 188–189 (1972)Google Scholar
  5. 5.
    Bingham, N.H.: Tauberian theorems and the central limit theorem. Ann. Probability 9, 221–231 (1981)Google Scholar
  6. 6.
    Bingham, N.H.: On Tauberian theorems in probability theory. Preprint, Westfield Coll. Univ. of London (1982)Google Scholar
  7. 7.
    Chow, Y.S.: Delayed sums and Borel summability of independent, identically distributed random variables. Bull. Inst. Math. Acad. Sinica 1, 207–220 (1973)Google Scholar
  8. 8.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. 1. 3rd ed. New York: Wiley 1968Google Scholar
  9. 9.
    Gerber, H.U.: The discounted central limit theorem and its Berry-Esséen analogue. Ann. Math. Statist. 42, 389–392 (1971)Google Scholar
  10. 10.
    Hardy, G.H.: Divergent Series. Oxford University Press 1967Google Scholar
  11. 11.
    Heyde, C.C.: A nonuniform bound on convergence to normality. Ann. Probability 3, 903–907 (1975)Google Scholar
  12. 12.
    Lai, T.L.: Summability methods for independent, identically distributed random variables. Proc. Amer. Math. Soc. 45, 253–261 (1974)Google Scholar
  13. 13.
    Maejima, M.: A nonuniform estimate in the central limit theorem for m-dependent random variables. Keio Engrg. Rep. 31, 15–20 (1978)Google Scholar
  14. 14.
    Maejima, M.: Nonuniform estimates in the central limit theorem. Yokohama Math. J. 26, 137–149 (1978)Google Scholar
  15. 15.
    Maejima, M.: A note on the nonuniform rate of convergence to normality. Yokohama Math. J. 28, 97–106 (1980)Google Scholar
  16. 16.
    Osipov, L.V.: Refinement of Lindeberg's theorem. Theory Probability Appl. 11, 299–302 (1966)Google Scholar
  17. 17.
    Petrov, V.V.: Sums of Independent Random Variables. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  18. 18.
    Pruitt, W.E.: Summability of independent random variables. J. Math. and Mech. 15, 769–776 (1966)Google Scholar
  19. 19.
    Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloquim Publications, Vol. 23, 1967Google Scholar
  20. 20.
    Watson, G.N.: The Theory of Bessel Functions. 2nd ed. Cambridge University Press 1966Google Scholar
  21. 21.
    Zeller, K., Beekmann, W.: Theorie der Limitierungsverfahren. Berlin-Heidelberg-New York: Springer 1970Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Paul Embrechts
    • 1
  • Makoto Maejima
    • 2
  1. 1.Department of MathematicsImperial CollegeLondonUK
  2. 2.Department of MathematicsKeio UniversityYokohamaJapan

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