Summary
We study partial sums of a stationary sequence of dependent random variables of the form \(W_n = \sum\limits_1^n {\xi \left( {S_k } \right)}\). Here S k =X 1 + ... +X k where the X i are i.i.d. integer valued, and ξ(n), n∈ℤ are also i.i.d. and independent of the X's. It is assumed that the X's and ξ's belong to the domains of attraction of different stable laws of indices 1<α≦2 and 0<β≦2. It is shown that for some δ>\(\frac{1}{2}\), n −δ W [nt] converges weakly as n→∞ to a self similar process with stationary increments, which depends on α and β. The constant δ is related to α and β via δ=1−α −1+(αβ)−1.
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Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Boylan, E.: Local times for a class of Markoff processes. Illinois J. Math. 8, 19–39 (1964)
Darling, D.A., Kac, M.: On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84, 444–458 (1957)
Dobrushin, R.L.: Gaussian and their subordinated self-similar random generalized fields. Ann. Probability 7, 1–28 (1979)
Dobrushin, R.L., Major, P.: Non central limit theorems for non linear functionals of Gaussian fields. [Z. Wahrscheinlichkeitstheorie verw. Gebiete, to appear
Feller, W.: An introduction to probability theory and its applications. Vol. II, 2nd ed. N.Y.: Wiley 1971
Getoor, R.K., Kesten, H.: Continuity of local times for Markov processes. Compositio Math. 24, 277–303 (1972)
Gikhman, I.I., Skorokhod, A.V.: Introduction to the theory of random processes. Philadelphia: Saunders 1969
Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Reading: Addison-Wesley 1954
Ito, K.: Stochastic processes. Aarhus University Lecture Notes Series, No. 16, 1969
Kesten, H., Kozlov, M.V., Spitzer, F.: A limit law for random walk in a random environment. Compositio Math. 30, 145–168 (1975)
Lamperti, J.: Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78 (1962)
Meyer, P.A.: Un cours sur les integrales stochastiques, Séminaire de Probabilités X. Univ. de Strasbourg. Lecture Notes in Math. 511. Berlin-Heidelberg-New York: Springer 1976
Spitzer, F.L.: Principles of random walk. 2nd ed. N.Y.: Springer 1976
Stone, C.J.: On local and ratio limit theorems. Proc. 5-th Berkeley Sympos. Math. Statist. Probab. Univ. Calif. 217–224, 1966
Taqqu, M.: Convergence of integrated processes of arbitrary Hermite rank. [To appear in Z. Wahrscheinlichkeitstheorie verw. Gebiete]
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To Leo Schmetterer on his 60th anniversary
Supported by the NSF at Cornell University
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Kesten, H., Spitzer, F. A limit theorem related to a new class of self similar processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 50, 5–25 (1979). https://doi.org/10.1007/BF00535672
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DOI: https://doi.org/10.1007/BF00535672