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A characterization of stable processes

Published online by Cambridge University Press:  14 July 2016

Eugene Lukacs*
Affiliation:
Catholic University of America

Extract

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t1, t2 ɛ T, t1 〈 t2; the random variable X(t2) – X(t1) is called the increment of the process X(t) over the interval [t1, t2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

Type
Research Papers
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

[1] Cramer, H. (1947) Problems in probability theory. Ann. Math. Statist. 18, 165193.CrossRefGoogle Scholar
[2] LoèVe, M. (1963) Probability Theory. Third ed. D. van Nostrand, Princeton, N.J. Google Scholar
[3] Lukacs, E. (1968) Stochastic Convergence. Heath Mathematical Monographs. Lexington, Mass. Google Scholar
[4] Lukacs, E. (1967) Une caractérisation des processus stables et symmétriques. C.R. Acad. Sci. Paris 264, 959960.Google Scholar