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A remark on positive sojourn times of symmetric processes

Part of: Markov processes Limit theorems Distribution theory - Probability

Published online by Cambridge University Press:  28 March 2018

Christophe Profeta
Christophe Profeta*
Affiliation:
Université d'Évry-Val d'Essonne and CNRS
*
* Postal address: LaMME, Bâtiment I.B.G.B.I., 3ème étage, 23 Bd. de France, 91037 Evry Cedex, France. Email address: christophe.profeta@univ-evry.fr
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Abstract

We show that under some slight assumptions, the positive sojourn time of a product of symmetric processes converges towards ½ as the number of processes increases. Monotony properties are then exhibited in the case of symmetric stable processes, and used, via a recurrence relation, to obtain upper and lower bounds on the moments of the occupation time (in the first and third quadrants) for two-dimensional Brownian motion. Explicit values are also given for the second and third moments in the n-dimensional Brownian case.

Keywords

Sojourn timearcsine lawstrong law of large numbers

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60J65: Brownian motion 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 69 - 81
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Bingham, N. H. (1996). The strong arc-sine law in higher dimensions. In Convergence in Ergodic Theory and Probability, de Gruyter, Berlin, pp. 111–116. CrossRefGoogle Scholar
[2]Bingham, N. H. and Doney, R. A. (1988). On higher-dimensional analogues of the arc-sine law. J. Appl. Prob. 25, 120–131. CrossRefGoogle Scholar
[3]Bingham, N. H. and Rogers, L. C. G. (1991). Summability methods and almost-sure convergence. In Almost Everywhere Convergence II, Academic Press, Boston, MA, pp. 69–83. CrossRefGoogle Scholar
[4]Desbois, J. (2007). Occupation times for planar and higher dimensional Brownian motion. J. Phys. A 40, 2251–2262. CrossRefGoogle Scholar
[5]Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954). Tables of Integral Transforms, Vol. II. McGraw-Hill, New York. Google Scholar
[6]Ernst, P. A. and Shepp, L. (2017). On occupation times of the first and third quadrants for planar Brownian motion. J. Appl. Prob. 54, 337–342. CrossRefGoogle Scholar
[7]Getoor, R. K. and Sharpe, M. J. (1994). On the arc-sine laws for Lévy processes. J. Appl. Prob. 31, 76–89. CrossRefGoogle Scholar
[8]Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th edn. Elsevier, Amsterdam. Google Scholar
[9]Janson, S. (2010). Moments of gamma type and the Brownian supremum process area. Prob. Surveys 7, 1–52. Google Scholar
[10]Lyons, R. (1988). Strong laws of large numbers for weakly correlated random variables. Michigan Math. J. 35, 353–359. Google Scholar
[11]McKeanH. P., Jr. H. P., Jr. (1963). A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2, 227–235. Google Scholar
[12]Meyre, T. and Werner, W. (1995). On the occupation times of cones by Brownian motion. Prob. Theory Relat. Fields 101, 409–419. Google Scholar
[13]Mountford, T. S. (1990). Limiting behaviour of the occupation of wedges by complex Brownian motion. Prob. Theory Relat. Fields 84, 55–65. Google Scholar
[14]Nakayama, K. (1997). On the asymptotic behavior of the occupation time in cones of d-dimensional Brownian motion. Proc. Japan Acad. Ser. A Math. Sci. 73, 26–28. Google Scholar