Some Applications of Distribution Functions of Sequences

This expository paper presents some old and some new results on distribution functions of sequences xn ∈ [0, 1), n = 1, 2, . . . Firstly we describe old applications: Statistically independent sequences; statistically convergent se- quences; statistical limit points; and uniform maldistributed sequences. Then we give some recent results: Benford’s law; copulas; and ratio sequences. Secondly we present some methods for computing the set G(xn) of all distribution func- tions of xn: directly by definition of distribution functions; using connectivity 1 1 of G(xn); solving a moment problem X1 = 0 g(x)dx, X2 = 0 xg(x)dx and X3 = 0 1 g2(x)dx for distribution functions g(x); and mapping xn to f (xn), for some function f : [0, 1] → [0, 1]. Parts of this paper were presented at the UDT conferences in Marseilles 2008, Strobl 2010, Smolenice 2012, and Ostravice 2014 and also in MCQMC conference, Warszawa 2010.