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Liminf behaviours of the windings and Lévy's stochastic areas of planar Brownian motion

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Séminaire de Probabilités XXVIII

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References

  • Abramowitz, M. & Stegun, I.A. (1965). Handbook of Mathematical Functions. Dover, New York.

    MATH  Google Scholar 

  • Baldi, P. (1986). Large deviations and functional iterated law for diffusion processes. Probab. Th. Rel. Fields 71, 435–453.

    Article  MathSciNet  MATH  Google Scholar 

  • Bélisle, C. (1991). Windings of spherically symmetric random walks via Brownian embedding. Statist. Probab. Letters 12 345–349.

    Article  MathSciNet  MATH  Google Scholar 

  • Berthuet, R. (1981). Loi du logarithme itéré pour certaines intégrales stochastiques. Ann. Sci. Univ. Clermont-Ferrand II Math. 19 9–18.

    MathSciNet  MATH  Google Scholar 

  • Berthuet, R. (1986). Etude de processus généralisant l'Aire de Lévy. Probab. Th. Rel. Fields 73 463–480.

    Article  MathSciNet  MATH  Google Scholar 

  • Bertoin, J. & Werner, W. (1994a). Comportement asymptotique du nombre de tours effectués par la trajectoire brownienne plane. This volume.

    Google Scholar 

  • Bertoin, J. & Werner, W. (1994b). Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process. This volume.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Csáki, E. (1978). On the lower limits of maxima and minima of Wiener process and partial sums. Z. Wahrscheinlichkeitstheorie verw. Gebiete 43 205–221.

    Article  MathSciNet  MATH  Google Scholar 

  • Dorofeev, E.A. (1994). The central limit theorem for windings of Brownian motion and that of plane random walk. Preprint.

    Google Scholar 

  • Durrett, R. (1982). A new proof of Spitzer's result on the winding of two-dimensional Brownian motion. Ann. Probab. 10, 244–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Feller, W. (1951). The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statist. 22, 427–432.

    Article  MathSciNet  MATH  Google Scholar 

  • Franchi, J. (1993). Comportement asymptotique presque sûr des nombres de tours effectués par le mouvement brownien d'une variété riemannienne compacte de dimension 2 ou 3. Technical Report No. 189, Laboratoire de Probabilités Université Paris VI. April 1993.

    Google Scholar 

  • Gruet, J.-C. & Mountford, T.S. (1993). The rate of escape for pairs of windings on the Riemann sphere. Proc. London Math. Soc. 48, 552–564.

    Article  MathSciNet  MATH  Google Scholar 

  • Helmes, K. (1985). On Lévy's area process. In: Stochastic Differential Systems (Eds.: N. Christopeit, K. Helmes & M. Kohlmann). Lect. Notes Control Inform. Sci. 78 187–194, Springer, Berlin.

    Chapter  Google Scholar 

  • Helmes, K. (1986). The “local” law of the iterated logarithm for processes related to Lévy's stochastic area process. Stud. Math. 83, 229–237.

    MathSciNet  MATH  Google Scholar 

  • Itô, K. & McKean, H.P. (1974). Diffusion Processes and their Sample paths. 2nd Printing. Springer, Berlin.

    MATH  Google Scholar 

  • Kochen, S. & Stone, C. (1964). A note on the Borel-Cantelli lemma. Illinois. J. Math. 8, 248–251.

    MathSciNet  MATH  Google Scholar 

  • Ledoux, M. & Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin.

    Book  MATH  Google Scholar 

  • Lévy, P. (1951). Wiener's random function, and other Laplacian random functions. Proc. Second Berkeley Symp. Math. Statist. Probab. 171–181.

    Google Scholar 

  • Lipschutz, M. (1956). On strong bounds for sums of independent random variables which tend to a stable distribution. Trans. Amer. Math. Soc. 81, 135–154.

    Article  MathSciNet  MATH  Google Scholar 

  • Lyons, T. & McKean, H.P. (1984) Windings of the plane Brownian motion. Adv. Math. 51, 212–225.

    Article  MathSciNet  MATH  Google Scholar 

  • Messulam, P. & Yor, M. (1982). On D. Williams' pinching method and some applications. J. London Math. Soc. 26 348–364.

    Article  MathSciNet  MATH  Google Scholar 

  • Pitman, J.W. & Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheorie verw. Gebiete 59, 425–457.

    Article  MathSciNet  MATH  Google Scholar 

  • Pitman, J.W. & Yor, M. (1986). Asymptotic laws of planar Brownian motion. Ann. Probab. (Special Invited Paper) 14, 733–779.

    Article  MathSciNet  MATH  Google Scholar 

  • Pitman, J.W. & Yor, M. (1989). Further asymptotic laws of planar Brownian motion. Ann. Probab. 17 965–1011.

    Article  MathSciNet  MATH  Google Scholar 

  • Rogers, L.C.G. & Williams, D. (1987). Diffusions, Markov Processes and Martingales, vol. II: Itô Calculus. Wiley, Chichester.

    MATH  Google Scholar 

  • Shi, Z. (1994). Windings of Brownian motion and random walk in ℝ2. Preprint.

    Google Scholar 

  • Spitzer, F. (1958). Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc., 87 187–197.

    MathSciNet  MATH  Google Scholar 

  • Williams, D. (1974). A simple geometric proof of Spitzer's winding number formula for 2-dimensional Brownian motion. Univ. College, Swansea. Unpublished.

    Google Scholar 

  • Yor, M. (1992). Some Aspects of Brownian Motion. Part I: Some Special Functionals. Birkhäuser, Basel. *** DIRECT SUPPORT *** A00I6B40 00005

    MATH  Google Scholar 

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Jacques Azéma Marc Yor Paul André Meyer

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© 1994 Springer-Verlag

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Shi, Z. (1994). Liminf behaviours of the windings and Lévy's stochastic areas of planar Brownian motion. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXVIII. Lecture Notes in Mathematics, vol 1583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073841

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  • DOI: https://doi.org/10.1007/BFb0073841

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