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Finite difference equations and convergence rates in the central limit theorem

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EN
Abstrakty
EN
We apply the theory of finite difference equations to the central limit theorem, using interpolation of Banach spaces and Fourier multipliers. Let S*n be a normalized sum of i.i.d. random vectors, converging weakly to a standard normal vector N. When does ǁEg (x + S*n) -E g (X + N)ǁLp(dx)tend to zero at a specified rate? We show that, under moment conditions, membership of g in various Besov spaces is often sufficient and sometimes necessary. The results extend to signed probability.
Rocznik
Strony
153--166
Opis fizyczny
Bibliogr. 29 poz.
Twórcy
autor
  • Saab Systems, S:t Olofsgatan 9A, 75321 Uppsala, Sweden
Bibliografia
  • [1] A. D. Barbour, Asymptotic expansions based on smooth functions in the central limit theorem, Probab. Theory Related Fields 72 (1986), pp. 289-303.
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  • [3] P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
  • [4] R. P. Boas, Lipschitz behavior and integrability of characteristic functions, Ann. Math. Statist. 38 (1967), pp. 32-36.
  • [5] I. S. Borisov, D. A. Panchenko, and G. I. Skilyagina, On minimal smoothness conditions for asymptotic expansions of moments in the CLT Siberian Adv. Math. 8 (1998), pp. 80-95.
  • [6] F. Gotze and C. Hipp, Asymptotic expansions in the central limit theorem under moment conditions, Z. Wahrsch. Verw. Gebiete 42 (1978), pp. 67-87.
  • [7] G. W. Hedstrom, The rate of convergence of some difference schemes, SIAM J. Numer. Anal. 5 (1968), pp. 363-406.
  • [8] C. C. Heyde and T. Nakata, On the asymptotic equivalence of Lp metrics for convergence to normality, Z. Wahrsch. Verw. Gebiete 68 (1984), pp. 97-106.
  • [9] K. J. Hochberg, Central limit theorem for signed distributions, Proc. Amer. Math. Soc. 79 (1980), pp. 298-302.
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  • [11] L. Hormander, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer, New York 1990.
  • [12] I. A. Ibragimov, On the accuracy of Gaussian approximation to the distribution functions of sums of independent variables, Theory Probab. Appl. 11 (1966), pp. 559-579.
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  • [14] J. Lofstrom, Besov spaces in theory of approximation, Ann. Mat. Pura Appl. (4) 85 (1970), pp. 93-184.
  • [15] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Press, 1976.
  • [16] J. Peetre and V. Thom Ce, On the rate of convergence for discrete initial-value problems, Math. Scand. 21 (1967), pp. 159-176.
  • [17] E. J. G. Pitman, On the derivatives of a characteristic function at the origin, Ann. Math. Statist. 27 (1956), pp. 1156-1160.
  • [18] R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-value Problems, 2nd edition, Interscience, New York 1967.
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  • [20] T. L. Shervashidze, On the convergence of densities of sums of independent random vectors, Lecture Notes in Math. No 1021, Springer, New York 1982, pp. 576-586.
  • [21] G. Strang, Polynomial approximation of Bernstein type, Trans. Amer. Math. Soc. 105 (1962), pp. 525-535.
  • [22] Yu. P. Studnev, Some generalizations of limit theorems in probability theory, Theory Probab. Appl. 12 (1967), pp. 668-672.
  • [23] V. Thomée, Stability of difference schemes in the maximum-norm, J. Differential Equations 1 (1965), pp. 273-292.
  • [24] V. Thomée, Finite difference methods for linear parabolic equations, in: Handbook of Numerical Analysis I, North-Holland, Amsterdam 1990, pp. 5-196.
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  • [29] A. I. Zukov, A limit theorem for difference operators (in Russian), Uspekhi Mat. Nauk 14 (1959), pp. 129-1 36.
Typ dokumentu
Bibliografia
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