Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations

Sera, Toru; Yano, Kouji

doi:10.1090/tran/7755

Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations
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by Toru Sera and Kouji Yano PDF

Trans. Amer. Math. Soc. 372 (2019), 3191-3209 Request permission

Abstract:

We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti’s generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.

References

Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400, DOI 10.1090/surv/050
Martin Barlow, Jim Pitman, and Marc Yor, Une extension multidimensionnelle de la loi de l’arc sinus, Séminaire de Probabilités, XXIII, Lecture Notes in Math., vol. 1372, Springer, Berlin, 1989, pp. 294–314 (French). MR 1022918, DOI 10.1007/BFb0083980
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
John Lamperti, An occupation time theorem for a class of stochastic processes, Trans. Amer. Math. Soc. 88 (1958), 380–387. MR 94863, DOI 10.1090/S0002-9947-1958-0094863-X
Paul Lévy, Sur certains processus stochastiques homogènes, Compositio Math. 7 (1939), 283–339 (French). MR 919
Maximilian Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math. 37 (1980), no. 4, 303–314. MR 599464, DOI 10.1007/BF02788928
Maximilian Thaler, Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math. 46 (1983), no. 1-2, 67–96. MR 727023, DOI 10.1007/BF02760623
Maximilian Thaler, A limit theorem for sojourns near indifferent fixed points of one-dimensional maps, Ergodic Theory Dynam. Systems 22 (2002), no. 4, 1289–1312. MR 1926288, DOI 10.1017/S0143385702000573
M. Thaler and R. Zweimüller, Distributional limit theorems in infinite ergodic theory, Probab. Theory Related Fields 135 (2006), no. 1, 15–52. MR 2214150, DOI 10.1007/s00440-005-0454-3
Shinzo Watanabe, Generalized arc-sine laws for one-dimensional diffusion processes and random walks, Stochastic analysis (Ithaca, NY, 1993) Proc. Sympos. Pure Math., vol. 57, Amer. Math. Soc., Providence, RI, 1995, pp. 157–172. MR 1335470, DOI 10.1090/pspum/057/1335470
Yuko Yano, On the joint law of the occupation times for a diffusion process on multiray, J. Theoret. Probab. 30 (2017), no. 2, 490–509. MR 3647066, DOI 10.1007/s10959-015-0654-4
Roland Zweimüller, Infinite measure preserving transformations with compact first regeneration, J. Anal. Math. 103 (2007), 93–131. MR 2373265, DOI 10.1007/s11854-008-0003-y