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Central Limit Theorems for Exchangeable Random Variables When Limits Are Scale Mixtures of Normals

Author

Listed:
  • Xinxin Jiang

    (Rhodes College)

  • Marjorie G. Hahn

    (Tufts University)

Abstract

Central limit theorems for exchangeable random variables are studied when limits are scale mixtures of normals. First, necessary and sufficient conditions are given under the asymptotic tail probability condition for the mixands: $$nP^\omega \left\{ {\left| {\xi _1 } \right| >\varepsilon b_n } \right\}\xrightarrow{P}0.$$ Second, when the weak limits have a particular form, i.e., the mixing measure comes directly from de Finetti's Theorem, necessary and sufficient conditions are given. Finally, some applications are discussed.

Suggested Citation

  • Xinxin Jiang & Marjorie G. Hahn, 2003. "Central Limit Theorems for Exchangeable Random Variables When Limits Are Scale Mixtures of Normals," Journal of Theoretical Probability, Springer, vol. 16(3), pages 543-571, July.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:3:d:10.1023_a:1025612330587
    DOI: 10.1023/A:1025612330587
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    References listed on IDEAS

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    1. Hall, Joyce A. & Brorsen, B. Wade & Irwin, Scott H., 1989. "The Distribution of Futures Prices: A Test of the Stable Paretian and Mixture of Normals Hypotheses," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(1), pages 105-116, March.
    2. Cheng, Tsung-Lin & Chow, Yuan-Shih, 2002. "On stable convergence in the central limit theorem," Statistics & Probability Letters, Elsevier, vol. 57(4), pages 307-313, May.
    3. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    4. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
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