One Dimensional Case

Abstract

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).