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Proceedings of the American Mathematical Society

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A note on the central limit theorem for square-integrable processes


Author: Marjorie G. Hahn
Journal: Proc. Amer. Math. Soc. 64 (1977), 331-334
MSC: Primary 60F05
DOI: https://doi.org/10.1090/S0002-9939-1977-0448487-X
MathSciNet review: 0448487
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Abstract: A method is given for constructing sample-continuous processes which do not satisfy the central limit theorem in $ C[0,1]$. Let $ \{ X(t):t \in [0,1]\} $ be a stochastic process. Using our method we characterize all possible nonnegative functions f for which the condition

$\displaystyle E( X(t) - X(s) )^2 \leqslant f( \vert t - s \vert )$

alone is sufficient to imply that $ X(t)$ satisfies the central limit theorem in $ C[0,1]$.

References [Enhancements On Off] (What's this?)

  • [R] M. Dudley (1967), The sizes of compact subsets of Hilbert space and continuity of Guassian processes, J. Functional Analysis 1, 290-330. MR 0220340 (36:3405)
  • [M] G. Hahn (1977), Conditions for sample-continuity and the central limit theorem, Ann. Probability (to appear). MR 0440679 (55:13551)
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  • [J] P. Kahane (1968), Some random series of functions, Heath, Boston, Mass. MR 0254888 (40:8095)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0448487-X
Keywords: Central limit theorem, second-order processes, random Fourier series
Article copyright: © Copyright 1977 American Mathematical Society

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