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.

Managing pressure: patterns of appraisals and coping strategies of non-elite and elite athletes during competition

The purpose of this study was to compare moment-to-moment appraisals and coping strategies of 4 non-elite and 2 elite male trap shooters during competitions and in particular during periods of competition perceived as critical to performance. Appraisals and coping patterns of trap shooters were captured via verbal reports of thinking provided between sets of shots during major competitions. Verbal reports were coded according to an appraisal and coping typology. Coded data as well as shooting performance data were subjected to a sequential analysis of probabilities of pairs of events. Fewer reports of negative appraisals (NEGAs) and more frequent reports of problem-focused coping (PFC) were observed among both elite athletes compared to non-elite athletes. After making a NEGA, non-elite shooters often progressed to the next target without attempting to cope, whereas elite shooters used both PFC and emotion-focused coping (EFC) before proceeding to the next target. After missing a target, the non-elite athletes used more EFC than expected. These results indicate that elite athletes are more likely to cope with NEGAs than non-elite athletes using a wider variety of coping strategies. Athletes might benefit from increased awareness of the potentially detrimental impact of NEGAs on performance and by integrating coping strategies within preparatory routines.