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The characteristic function is a way to describe a random variable. The characteristic function, a function of t, determines the behavior and properties of the probability distribution of the random variable X.
A characterization is a certain distributional or statistical property of a statistic or statistics that uniquely determines the associated stochastic model. This chapter provides a brief survey of the huge literature on this topic.
The uniform distribution explained, with examples, solved exercises and detailed proofs of important results
Characterization A distribution function F : R → [0,1] satisfies the following properties (i) F is increasing. (ii) F(x) → 0 for x → −∞, F(x) → 1 for x → ∞. (iii) F is right continuous. Important characterization: Any function F : R → [0,1] satisfying the properties (i)-(iii) above is the distribution function for a unique ...
Abstract. We elaborate the notions of Martin-L ̈of and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform distribution of sequences.
technique for determining if a sequence of random vectors converges in distribution. It is based on a characterization of distributions by something simpler than the means of all bounded continuous functions. The means of a special collection of bounded continuous functions, namely {exp(it⊤x) : t ∈ IRp}, are enough to characterize a ...
Topics discussed include uniform distribution characterization of Benford's law, uniform distribution of sequences and functions, and uniform distribution of random variables.
