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A further generalization of the arc-sine law

Published online by Cambridge University Press:  09 April 2009

C. C. Heyde
C. C. Heyde
Affiliation:
University of Sheffield
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Let Xi, i = 1,2,3,… be a sequence of independent and identically distributed random variables and write S0 = 0, Sn = ∑ni=1Xi, n ≧ 1. Let In(0), In(1), …, In (n) be that unique permutaion of 1, 2, …, n such that SIn(0) ≦ SIn(1) ≦ … ≦ SIn(n) and such that if Si = Sk with i < k then In(k) < In(j). Thus, In(j) is an index of the j-th largest partial sum.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 2 , May 1968 , pp. 369 - 372
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Darling, D. A., ‘Sums of symmetrical random variables’, Proc. Amer. Math. Soc. 2 (1951), 511–517.CrossRefGoogle Scholar
[2]Heyde, C. C., ‘Some local limit results in fluctuation theory’, J. Austral. Math. Soc. 7 (1967), 455–464.CrossRefGoogle Scholar
[3]Spitzer, F. L., ‘A combinatorial lemma and its application to probability theory’, Trans. Amer. Math. Soc. 82 (1956), 232–339.CrossRefGoogle Scholar