Factorial moment generating function

Abstract: We introduce a class of two-parameter discrete dispersion models, obtained by combining convolution with a factorial tilting operation, similar to exponential dispersion models which combine convolution and exponential tilting. The equidispersed Poisson model has a special place in this approach, whereas several overdispersed discrete distributions, such as the Neyman Type A, Polya–Aeppli, negative binomial and Poisson-inverse Gaussian, turn out to be Poisson–Tweedie factorial dispersion models with power dispersion functions, analogous to ordinary Tweedie exponential dispersion models with power variance functions. Using the factorial cumulant generating function as tool, we introduce a dilation operation as a discrete analogue of scaling, generalizing binomial thinning. The Poisson–Tweedie factorial dispersion models are closed under dilation, which in turn leads to a Poisson–Tweedie asymptotic framework where Poisson–Tweedie models appear as dilation limits. This unifies many discrete convergence results and leads to Poisson and Hermite convergence results, similar to the law of large numbers and the central limit theorem, respectively. The dilation operator also leads to a duality transformation which in some cases transforms overdispersion into underdispersion and vice versa. Finally, we consider the multivariate factorial cumulant generating function, and introduce a multivariate notion of over- and underdispersion, and a multivariate zero inflation index.

Abstract: We discuss the two standard constructions used in the search for intermittency, the exclusive and inclusive scaled factorial moments. We propose the use of a new scaled factorial moment that reduces to the exclusive moment in the appropriate limit and is free of undesirable multiplicity correlations that are contained in the inclusive moment. We show that there are some similarities among most of the models that have been proposed to explain factorial-moment data, and that these similarities can be used to increase the efficiency of testing these models. We begin by calculating factorial moments from a simple independent-cluster model that assumes only approximate boost invariance of the cluster rapidity distribution and an approximate relation among the moments of the cluster multiplicity distribution. We find two scaling laws that are essentially model independent. The first scaling law relates the moments to each other with a simple formula, indicating that the different factorial moments are not independent. The second scaling law relates samples with different rapidity densities. We find evidence for much larger clusters in heavy-ion data than in light-ion data, indicating possible {ital spatial} intermittency in the heavy-ion events.

Abstract: In this paper we generalize the concept of sub-Gaussian random variable to that of “locally” sub-Gaussian random variable. Some properties of locally sub-Gaussian random variables are presented. It is shown that a “local” version of the moment inequality used by Taylor and Hu in 1987 can be used to give an equally simple proof of the strong law of large numbers for locally sub-Gaussian random variables.