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Strong law of large numbers for the middle part of a sample

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Abstract

Let {Xn} n=1 be a sequence of independent, symmetric random variables and let {Xin} i=1 n be the absolute order statistics. The rate of growth of\(\sum\limits_{i = 1}^{n - 2} {X_{i,n} }\) and X2,n is investigated for n→∞.

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Literature cited

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    V. A. Egorov, “On the central limit theorem in the absence of extremal absolute order statistics,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,142, 59–67 (1985).

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    V. A. Egorov and V. B. Nevzorov, “On the rate of convergence to the normal law of linear combinations of absolute order statistics,” Teor. Veroyatn. Primen.,20, No. 1, 207–215 (1975).

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    V. A. Egorov, “On the strong law of large numbers and the law of the iterated logarithm for a sequence of independent random variables,” Teor. Veroyatn. Primen.,15, 520–527 (1970).

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    V. A. Egorov, “On an approach to the proof of the theorems on the law of the iterated logarithm,” Teor. Veroyatn. Primen.,29, No. 1, 125–132 (1984).

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    H. A. David, Order Statistics, Wiley, New York (1970).

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 25–31, 1988.

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Egorov, V.A. Strong law of large numbers for the middle part of a sample. J Math Sci 52, 2878–2883 (1990). https://doi.org/10.1007/BF01103741

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Keywords

  • Middle Part
  • Order Statistic
  • Absolute Order
  • Symmetric Random Variable