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The Bounds Based on the Functions of Observations for Maximum of Stable Law

Published online by Cambridge University Press:  20 November 2018

A. K. Basu  and
M. T. Wasan
A. K. Basu
Affiliation:
Queen's University, Kingston, Ontario
M. T. Wasan
Affiliation:
Queen's University, Kingston, Ontario
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Gnedenko and Kolmogorov [3, pp. 181-182] have shown that if Xn with law F(x) belong to the domain of normal attraction of a stable law of index 0<α<2, i.e. if partial sum Sn/an1/α converges in distribution to some stable law Vα, a>0 then there exist c1 and c2 such that

and

Type
Mathematical Notes
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 285 - 287
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Doubleday, W. G., and Wasan, M. T., A note on the maximum of the time distribution of standard Brownian motion, Sankhyá, Ser. A, 32 (1970), 347-349.Google Scholar
2. Gnedenko, B. V., Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math. 44 (1943), 423-453.Google Scholar
3. Gnedenko, B. V. and Kolmogorov, A. N., Limit distributions for sums of independent random variables, Addison-Wesley, Reading, Mass., 1954.Google Scholar