Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T14:03:22.444Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Functional Stable Limit Theorems

Published online by Cambridge University Press:  20 November 2018

D. R. Beuerman
D. R. Beuerman*
Affiliation:
Queen's University, Kingston, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,

1

where

and

where L(n) is a function of slow variation; also take S0=0, B0=l.

In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 2 , June 1973 , pp. 173 - 177
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Apostol, T.M., Mathematical analysis, Addison-Wesley, Reading, Mass., 1957.Google Scholar
2. Beuerman, D.R., On the limit distribution of the time of first passage over a curvilinear boundary, (Abstract), Canad. Math. Bull. (5) 12 (1969), p. 694.Google Scholar
3. Billingsley, P., Convergence of probability measures, Wiley, New York, 1968.Google Scholar
4. Breiman, L., Probability, Addison-Wesley, Reading, Mass., 1968.Google Scholar
5. Daniels, H.E., The probability distribution of the extent of a random chain, Proc. Cambridge Philos. Soc. 37 (1941), 244–261.Google Scholar
6. Feller, W., The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Stat. 22 (1951), 427–432.Google Scholar
7. Lukacs, E., Characteristic functions, Griffin and Co., London, 1960.Google Scholar
8. Siegmund, D., On the asymptotic normality of one-sided stopping rules, Ann. Math. Stat. 39 (1968), p. 1493.Google Scholar
9. Skorokhod, A.V., Limit theorems for stochastic processes with independent increments, Theor. Probability Appl. 2 (1957), 138–171.Google Scholar