A Functional LIL for d-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes

Springer Science and Business Media LLC - 2007
Joshua Rushton1
1University of Wisconsin, Madison, USA

Tóm tắt

We establish a functional LIL for the maximal process M(t) :=sup 0≤s≤t ‖X(s)‖ of an ℝ d -valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X.

Từ khóa


Tài liệu tham khảo

Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York (1980)

Bertoin, J.: Lévy Processes, vol. 121. Cambridge University Press, Cambridge (1996)

Byczkowski, T., Nolan, J.P., Rajput, B.: Approximation of multidimensional stable densities. J. Multivar. Anal. 46(1), 13–31 (1993)

Chen, X., Kuelbs, J., Li, W.: A functional LIL for symmetric stable processes. Ann. Probab. 28(1), 258–276 (2000)

Chung, K.L.: On the maximum partial sums of sequences of independent random variables. Trans. Am. Math. Soc. 64, 205–233 (1948)

de Acosta, A.: Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Probab. 11(1), 78–101 (1983)

de la Peña, V.H., Giné, E.: Decoupling. Springer, New York (1999)

Ethier, S.N., Kurtz, T.G.: Markov Processes. Wiley Inc., New York (1986)

Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley Inc., New York (1971)

Mogul’skiĭ, A.A.: Small deviations in the space of trajectories. Teor. Verojatnost. i Primenen. 19, 755–765 (1974)

Montgomery-Smith, S.J.: Comparison of sums of independent identically distributed random vectors. Probab. Math. Stat. 14(2), 281–285 (1994)

Rushton, J.: A Chung-type functional LIL for α-stable Lévy processes, with related invariance results. PhD thesis, University of Wisconsin (2005)

Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)

Seneta, E.: Regularly Varying Functions. Lecture Notes in Mathematics, vol. 508. Springer, Berlin (1976)

Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3, 211–226 (1964)

Wichura, M.: A functional form of Chung’s law of the iterated logarithm for maximum absolute partial sums. Unpublished Manuscript, 1973

Zolotarev, V.M.: One-Dimensional Stable Distributions, vol. 65. American Mathematical Society, Providence (1986). Translated from the Russian by H.H. McFaden, Translation edited by Ben Silver

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Thông tin tác giả