Eye movement dynamics and cognitive self-organization in typical and atypical development

This weak multiplicative distortion may be expressed in a couple of ways depending on how the distributions are instantiated. Exponential distributions can be generated by the sum of the squares of multiple independent normally distributed variables (i.e., the sum of independent components interacting with themselves, that is, accentuating their own contributions). Gamma distributions are generated by the sum of relatively few exponential distributions; exponential distributions are themselves specific cases of gamma distributions. Gamma distributions may also be generated by dividing the basic exponential distribution function by the gamma function, a continuous generalization of the factorial function (i.e., x!), itself a multiplicative transformation. As the number of added exponentials increases towards large limits (i.e., towards the limit of "very many random numbers" as noted in the main text), the gamma distribution converges towards the normal distribution. Alternatively, exponential distributions may be considered as the negative logarithm of uniform distributions in the range (0, 1). In this light, the sum of relatively few exponentials composing the gamma distribution is a logarithm of the product of relatively few uniformly distributed random variables. Because logarithm of repeated multiplication reduces to repeated addition of logarithms, the convergence of gamma distributions to normal distributions holds just the same. Although the values being added affect some aspects of the distribution, in the limit of very large sets of random variables or repeated additions, the distribution will converge to a normal distribution. Thus, in either the gamma or exponential case, no matter how multiplicativity appears in the generation of the distributions, it is quashed by repeated addition of very many random variables (whether those variables are exponential or squared normal). That is, skewed distributions following exponential or gamma form will collapse back towards normal distributions and thus reflect much stronger additive than multiplicative relationships among random variables. For this reason, we do not distinguish whether gamma or exponential distributions reflect greater or lesser multiplicativity: gamma distributions may be more multiplicative in general than in the limiting case of the exponential distribution if only because they involve dividing probabilities by values of the gamma function different from 1 (i.e., different from the multiplicative identity), but their convergence to normal distributions suggests that this multiplicativity is ultimately negligible in the largest limits.