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Weak convergence of the adjusted range of cumulative sums of exchangeable random variables

Published online by Cambridge University Press:  14 July 2016

Brent M. Troutman
Brent M. Troutman*
Affiliation:
U. S. Geological Survey
*
∗ Postal address: U.S. Geological Survey, Denver Federal Center, Box 25046, MS 420, Denver, CO 80225, U.S.A.
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Abstract

Let be the adjusted range of the cumulative sums of a sequence , where . Weak convergence results for random functions constructed from cumulative sums of {Xs} are used to obtain the asymptotic distribution and moments of when {Xs} are exchangeable, or symmetrically dependent, random variables.

Keywords

FUNCTIONAL CENTRAL LIMIT THEOREMEXCHANGEABLE SEQUENCES
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 297 - 304
Copyright
Copyright © Applied Probability Trust 1983 

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References

Anis, A. A. and Lloyd, E. H. (1975) Skew inputs and the Hurst effect. J. Hydrology 26, 39–54.CrossRefGoogle Scholar
Anis, A. A. and Lloyd, E. H. (1976) The expected value of the adjusted rescaled Hurst range of independent normal summands. Biometrika 63, 111–116.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Blum, J. R., Chernoff, H., Rosenblatt, M. and Teicher, H. (1958) Central limit theorems for interchangeable processes. Canad. J. Math. 10, 222–229.Google Scholar
Boes, D. C. and Salas-La Cruz, J. D. (1973) On the expected range and expected adjusted range of partial sums of exchangeable random variables. J. Appl. Prob. 10, 671–677.Google Scholar
De Finetti, B. (1937) La prévision: ses lois logiques, ses sources subjective. Ann. Inst. H. Poincaré 7, 1–68. (English translation in Studies in Subjective Probability (1964), ed. Kyburg, H. E. and Smokier, H. E.)Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Feller, W. (1951) The asymptotic distribtion of the range of sums of independent random variables. Ann. Math. Statist. 22, 427–432.CrossRefGoogle Scholar
Loeve, M. (1963) Probability Theory, 3rd edn. Van Nostrand, Princeton, NJ.Google Scholar
Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar
Troutman, B. M. (1978) Reservoir storage with dependent, periodic net inputs. Water Resources Res. 14, 395–401.Google Scholar
Troutman, B. M. (1979) Some results in periodic autoregression. Biometrika 66, 219–228.Google Scholar
Whitt, W. (1972) Complements to heavy traffic limit theorems for the GI/G/1 queue. J. Appl. Prob. 9, 185–191.Google Scholar