Summary
Let B be a 1-dimensional Brownian motion. In this paper ratios of the form A + (t)/A - (t), where A + is the (0, ∞)-occupation time functional of B and A -is a local time integral of an infinite (but locally finite) measure m with support in (-∞, 0), are studied. Conditions on m are given which ensure that such a ratio will be unbounded a.s. (or go to zero a.s.) as t»∞.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bojanic, R., Seneta, E.: Slowly varying functions and asymptotic relations. J. Math. Anal. Appl. 34, 302–315 (1971)
Erickson, K.B.: The strong law of large numbers when the mean is undefined. Trans. Am. Math. Soc. 185, 371–381 (1973)
Feller, W.: An introduction to probability theory and its applications II. New York: Wiley 1971
Fernique, X.M.: Integrabilité des vecteurs Gaussien. C.R. Acad. Sci. Paris Ser. A, 270, 1698–1699 (1970)
Ito, K., McKean, H.P., Jr.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1974
Kesten, H.: The limit points of a normalized random walk. Ann. Math. Stat. 41, 1173–1205 (1970)
London, R.R., McKean, H.P., Rogers, L.C.G., Williams, D.: A martingale approach to some Wiener-Hopf problems, I. Séminaire de Probabilities XVI. Lecture Notes in Math. 920, 41–67. Berlin-Heidelberg-New York: Springer 1982
Ray, D.: Sojourn times of diffusion process. Ill. J. Math. 7, 615–630 (1963)
Williams, D.: Decomposing the Brownian path. Bull. Am. Math. Soc. 76, 871–873 (1970)
Author information
Authors and Affiliations
Additional information
The work was supported in part by a grant from the National Science Foundation.
Rights and permissions
About this article
Cite this article
Erickson, K.B. A ratio ergodic theorem for increasing additive functionals. Probab. Th. Rel. Fields 72, 493–504 (1986). https://doi.org/10.1007/BF00344717
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00344717