Abstract
A functional law of the iterated logarithm is obtained for processes given by certain stochastic integrals. This extends earlier results by Shi(12) and Rémillard(10) who established analogues of the classical limit results of Chung(4) for a variety of processes, including Lévy’s stochastic area process. The functional aspects of our results are motivated by a paper of Wichura(13) on Brownian motion. Proofs depend on small ball probability estimates, and yield the small ball probabilities of the weighted sup-norm for the processes given by these stochastic integrals.
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References
S. Axler (1996) Linear Algebra Done Right Springer New York
N.H. Bingham C.M. Goldie J.L. Teugles (1987) Regular Variation Cambridge University Press Cambridge
X. Chen J. Kuelbs W. Li (2000) ArticleTitleA functional LIL for symmetric stable processes Ann. of Probab 28 258–276
K.L. Chung (1948) ArticleTitleOn the maximum partial sums of sequences of independent random variables Trans. Amer. Math. Soc 64 205–233
M. Csörgő P. Révész (1981) Strong Approximations in Probability and Statistics Academic Press New York
N. Ikeda S. Watanabe (1989) Stochastic Differential Equations and Diffusion Processes North-Holland Publishing Company Amsterdam
S. Janson M. Wichura (1983) ArticleTitleInvariance principles for stochastic area and related stochastic integrals. Stoch Proc. Applic 16 71–84
J. Kuelbs W.V. Li (2002) ArticleTitleA functional LIL and some weighted occupation measure results for fractional Brownian motion J. Theoret. Probab 15 1007–1030
W.V. Li M.Q. Shao (2001) Gaussian processes : Inequalities, small ball probabilities, and applications, Stochastic processes: Theory and Methods Handbook of Statistics NumberInSeries19 North-Holland Publishing Company Amsterdam
B. Rémillard (1994) ArticleTitleOn Chung’s law of the iterated logarithm for some stochastic integrals Ann. Probab 22 1794–1802
L.C.G. Rogers D. Williams (1987) Diffusions, Markov Processes and Martingales Vol.II Ito Calculus Wiley New York.
Z. Shi (1994) Liminf behaviors of the windings and Lévy’s stochastic areas of planar Brownian motion J. Azéma P.A. Meyer M. Yor (Eds) Séminaire de Probabilitiés XXVIII Lectures Notes in Mathematics NumberInSeries1583 Springer Berlin
Wichura M. (1973), A functional form of Chung’s law of the iterated logarithm for maximum absolute partial sums, unpublished.
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Kuelbs, J., Li, W. A Functional LIL for Stochastic Integrals and the Lévy Area Process. J Theor Probab 18, 261–290 (2005). https://doi.org/10.1007/s10959-003-2604-9
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DOI: https://doi.org/10.1007/s10959-003-2604-9