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Sample function behavior of increasing processes of class L
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  • Published: September 1996

Sample function behavior of increasing processes of class L

  • Toshiro Watanabe1 

Probability Theory and Related Fields volume 104, pages 349–374 (1996)Cite this article

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Summary

We consider increasing processes {X(t)∶t≧0} of classL, that is, increasing self-similar processes with inswpendent increments. Leth(t) be an increasing positive function on (0,∞) withh(0+)=0 andh(∞)=∞. By virtue of the zero-one laws, there existsc (resp.C) ∈[0,∞] such that lim inf (resp. lim sup)X(t)/h(t)=c (resp.C) a.s. both ast tends to 0 and ast tends to ∞. We decide a necessary and sufficient condition for the existence ofh(t) withc orC=1 and explicitly constructh(t) in caseh(t) exists withc orC=1. Moreover, we give a criterion to classify functionsh(t) withc (orC)=0 andh(t) withc (orC)=∞ in caseh(t) does not exist withc (orC)=1.

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References

  1. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge: Cambridge University Press 1987

    Google Scholar 

  2. Cline, D.B.H.: Intermediate regular and II variation. Proc. London Math. Soc.68, 594–616 (1994)

    Google Scholar 

  3. Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete49, 335–347 (1979)

    Google Scholar 

  4. Fristedt, B.E.: The behavior of increasing stable processes for both small and large times. J. Math. Mech.13, 849–856 (1964)

    Google Scholar 

  5. Fristedt, B.E.: Sample function behavior of increasing processes with stationary, independent increments. Pac. J. Math.21, 21–33 (1967)

    Google Scholar 

  6. Fristedt, B.E.: Sample functions of stochastic processes with stationary, independent increments. In: Ney, P., Port, S. (eds.) Advances in Probability, Vol. 3, pp. 241–396. Marcel Dekker 1974

  7. Fristedt, B.E., Pruitt, W.E.: Lower functions for increasing random walks and subordinators. Z. Wahrsch. Verw. Gebiete18, 167–182 (1971)

    Google Scholar 

  8. Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Reading, MA: Addison-Wesley 1954

    Google Scholar 

  9. de Haan, L., Stadtmüller, U.: Dominated variation and related concepts and Tauberian theorems for Laplace transform. J. Math. Anal. Appl.108, 344–365 (1985)

    Google Scholar 

  10. Sato, K.: A note on infinitely divisible distributions and their Lévy measures. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A12, 101–109 (1973)

    Google Scholar 

  11. Sato, K.: ClassL of multivariate distributions and its subclasses. J. Multivariate Anal.10, 207–232 (1980)

    Google Scholar 

  12. Sato, K.: Self-similar processes with independent increments. Probab. Theory. Relat. Fields89, 285–300 (1991)

    Google Scholar 

  13. Sato, K., Yamazato, M.: On distribution functions of classL. Z. Wahrsch. Verw. Gebiete43, 273–308 (1978)

    Google Scholar 

  14. Seneta, E.: Regularly varying functions. (Lect. Notes Math., Vol. 508) Berlin New York Heidelberg: Springer 1976

    Google Scholar 

  15. Steutel, F.W.: Preservation of Infinite Divisibility under Mixing and Related Topics. Math. Centre Tracts, No. 33, Amsterdam, 1970

  16. Wolfe, S.J.: On the unimodality ofL functions. Ann. Math. Statist.42, 912–918 (1971)

    Google Scholar 

  17. Yamazato, M.: Unimodality of infinitely divisible distribution functions of classL. Ann. Probab.6, 523–531 (1978)

    Google Scholar 

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Authors and Affiliations

  1. Center for Mathematical Sciences, The University of Aizu, 965, Aizu-Wakamatsu, Fukushima, Japan

    Toshiro Watanabe

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  1. Toshiro Watanabe
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Watanabe, T. Sample function behavior of increasing processes of class L. Probab. Th. Rel. Fields 104, 349–374 (1996). https://doi.org/10.1007/BF01213685

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  • Received: 12 April 1995

  • Revised: 08 September 1995

  • Issue Date: September 1996

  • DOI: https://doi.org/10.1007/BF01213685

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Mathematics Subject Classification (1991)

  • 60G18
  • 60J30
  • 60G17
  • 60E07
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