One Dimensional Case

Part of the Lecture Notes in Statistics book series (LNS, volume 71)


This chapter considers physical systems with one degree of freedom. The fractional part of the product of a large real number t and a random variable X, (tX)(mod 1), is studied in detail in Sect. 3.1. The random variable (tX)(mod 1) converges in the weak-star topology to a distribution uniform on [0, 1] if and only if the characteristic function of X vanishes at infinity (Theorem 3.2, Kemperman). A necessary and sufficient condition for convergence in the variation distance is that X have a density (Theorem 5.3). Convergence may take place at very slow rates (Proposition 3.8). Yet if the density of X has bounded variation, convergence is at a rate at least linear in t−1 (Theorem 3.9, Kemperman). Convergence may take place at faster rates for specific families of random variables (Proposition 3.11). Furthermore, there exists a family of random variables for which (tX)(mod 1) is identically uniform once t passes a certain threshold (Theorem 3.14).


Characteristic Function Lyapunov Exponent Fourier Coefficient Variation Distance Joint Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Department of EconomicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Departamento de Ingeniería IndustrialUniversidad de ChileSantiagoChile

Personalised recommendations