Asymptotic and structural properties of special cases of the Wright function arising in probability theory

This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function 1Ψ1(ρ, k; ρ, 0; x) = X∞ n=0 Γ(k + ρn) Γ(ρn) x n n! (|x| < ∞) when the parameter ρ ∈ (−1, 0)∪(0, ∞) and the argument x is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter k is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of ρ. The asymptotics of 1Ψ1(ρ, k; ρ, 0; x) are obtained under numerous assumptions on the behavior of the arguments k and x when the parameter ρ is both positive and negative. We also provide some integral representations and structural properties involving the ‘reduced’ Wright function 0Ψ1(−−; ρ, 0; x) with ρ ∈ (−1, 0) ∪ (0, ∞), which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions 0Ψ1(−−;±ρ, 0; ·) and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.

Big data: appearance and disappearance

The mainstay of Big Data is prediction in that it allows practitioners, researchers, and policy analysts to predict trends based upon the analysis of large and varied sources of data. These can range from changing social and political opinions, patterns in crimes, and consumer behaviour. Big Data has therefore shifted the criterion of success in science from causal explanations to predictive modelling and simulation. The 19th-century science sought to capture phenomena and seek to show the appearance of it through causal mechanisms while 20th-century science attempted to save the appearance and relinquish causal explanations. Now 21st-century science in the form of Big Data is concerned with the prediction of appearances and nothing more. However, this pulls social science back in the direction of a more rule- or law-governed reality model of science and away from a consideration of the internal nature of rules in relation to various practices. In effect Big Data offers us no more than a world of surface appearance and in doing so it makes disappear any context-specific conceptual sensitivity.