Summary
This paper considers the problem of the existence of set-indexed Lévy processes having regular sample paths defined over as large a class, A, as possible of subsets of the unit cube in ℝd. Regular sample paths means here the natural generalization of right continuity and left limits, to concepts of outer continuity and inner limits. A general integral condition involving the Lévy measure and the entropy exp(H(δ)) of the class A is obtained that is sufficient for the existence of such regular processes. In the particular case where the process is stable of index α, α∈(1, 2), the condition becomes
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Adler, R.J., Feigin, P.: On the cadlaguity of random measures. Ann. Probability 12, to appear 1984
Adler, R.J., Monrad, D., Scissors, R.H., Wilson, R.J.: Representations, decompositions and sample function continuity of random fields with independent increments. Stoch. Proc. Appl. 15, 3–30 (1983)
Bass, R.F., Pyke, R.: The space D(A) and weak convergence for set-indexed processes. To appear 1984
Beněs, V.: Characterization and decomposition of stochastic processes with stationary independent increments. (Abstract) Amer. Math. Soc. Bulletin 5, 246–247 (1958)
Bennett, G.: Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57, 33–45 (1962)
Chentsov, N.N.: Wiener random fields depending on several parameters. Dokl. Akad. Nauk SSSR (N.S.) 106, 607–609 (1956)
De Finetti, B.: Sulle funzioni a incremento aleatorio. Rend. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. 10, 163–168, 325–329, 548–553 (1929)
Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probability 1, 66–103 (1973)
Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10, 227–236 (1974)
Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probability 6, 899–929 (1978)
Dudley, R.M.: Lower layers in R 2 and convex sets in R 3 are not GB classes. Lecture Notes in Mathematics 700, 97–102. Berlin-Heidelberg-New York: Springer 1979
Ferguson, T.S., Klass, M.J.: A representation of independent increment processes without Gaussian components. Ann. Math. Statist. 43, 1634–1643 (1973)
Fristedt, B.: Sample functions of stochastic processes with stationary, independent increments. In: Advances in Probability and Related Topics, Vol. 3, pp. 241–396, New York: Dekker 1974
Ito, K.: On stochastic processes (I) (Infinitely divisible processes). Jap. J. Math. 18, 261–301 (1942)
Kallenberg, O.: Series of random processes without discontinuities of the second kind. Ann. Probability 2, 729–737 (1974)
Kingman, J.F.C.: Completely random measures. Pac. J. Math. 21, 59–78 (1967)
Lévy, P.: Théorie de l'addition des variables aléatoires. Paris: Gauthier-Villars 1937
Lévy, P.: Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars 1947
Pyke, R.: Partial sums of matrix arrays, and Brownian Sheets. In: Stochastic Analysis, ed. D.G. Kendall and E.F. Harding, pp. 331–348, London: Wiley 1973
Pyke, R.: A uniform central limit theorem for partial-sum processes. In: Probability, Statistics and Analysis. London Math. Soc. Lecture Note Series No. 79, 219–240 (1983)
Rossberg, H.-J., Jesiak, B., Siegel, G.: Continuation of distribution functions. In: Contributions to Probability: A collection of papers dedicated to Eugene Lukacs, ed. J. Gani and V.K. Rohatgi, pp. 29–48. New York: Academic Press 1981
Sato, K.: A note on infinitely divisible distributions and their Lévy measures. Sci. Rep. Tokyo Kyoiku Daigaku A 12, 101–109 (1973)
Taylor, S.J.: Sample path properties of processes with stationary, independent increments, In: Stochastic Analysis, ed. D.G. Kendall and E.F. Harding, pp. 387–414. London: Wiley 1973
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This work was partially supported by NSF Grants MCS-82-02861 and MCS-83-00581
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Bass, R.F., Pyke, R. The existence of set-indexed Lévy processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 66, 157–172 (1984). https://doi.org/10.1007/BF00531526
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DOI: https://doi.org/10.1007/BF00531526