Lower Bounds for the Distribution of Suprema of Brownian Increments and Brownian Motion Normalized by the Corresponding Modulus Functions

详细信息    查看全文

• 作者：Vladimir Dobric ; Lisa Marano
• 关键词：Brownian motion ; Global and local moduli of continuity of Brownian motion ; Lévy–Ciesielski construction of Brownian motion ; Law of the iterated logarithm
• 刊名：Journal of Theoretical Probability
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：29
• 期：1
• 页码：1-31
• 全文大小：542 KB
• 参考文献：1.Dobric, V. and Marano, L.: Rates of convergence for L évy’s modulus of continuity and H̆inc̆in’s law of the iterated logarithm. In: Hoffmann-Jørgensen, J., Marcus, M., Wellner, J. (eds.) High Dimensional Probability III, Progress in Probability, vol. 55, pp. 105–109 Birkh äuser, Basel (2003)
  2.Einmahl, U.: The Darling–Erdös theorem for sums of i.i.d. random variables. Probab. Theory Relat. Fields 82(2), 241–257 (1989)CrossRef MathSciNet MATH
  3.Erdös, P.: On the law of the iterated logarithm. Ann. Math. 43, 419–436 (1942)CrossRef MATH
  4.Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge (1954)MATH
  5.Kahane, J.P.: Some Random Series of Functions. Cambridge University Press, Cambridge (1985)MATH
  6.Khoshnevisan, D., Levin, D., Shi, Z.: Extreme-value analysis of the LIL for Brownian motion. Electron. Commun. Probab. Paper 20. 10, 196–206 (2005)
  7.Lévy, P.: Théorie de l’Additiondes Variables Alé atoires. Gauthier-Villars, Paris (1937)
  8.Meyer, Y.: Ondelettes et Opérateurs. Hermann, Paris (1990)
  9.Pinsky, M.A.: Brownian continuity modulus via series expansions. J. Theor. Probab. 14(1), 261–266 (2001)CrossRef MathSciNet MATH
  10.Steele, M.: Stochastic Calculus and Financial Applications (Stochastic Modelling and Applied Probability). Springer, Berlin (2001)CrossRef
  11.Talagrand, M., Ledoux, M.: Probability in Banach Spaces. Springer, Berlin (1980)
  
• 作者单位：Vladimir Dobric (1)
  Lisa Marano (2)
  
  1. Lehigh University, Bethlehem, PA, USA
  2. West Chester University of Pennsylvania, West Chester, PA, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

The Lévy–Ciesielski construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process \((W_{t})_{t\in \left[ 0,T\right] }\) normalized by the global modulus function, for all positive \(\varepsilon \) and \(\delta \). Additionally, uniform results over \(\delta \) are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in establishing the results for the standard Brownian motion case.