### Variance functions and Taylor’s law of fluctuation scaling

ABSTRACT: The variance function of a family of nonnegative random variables (rvs) gives the variance of each rv as a function of its mean, assuming each rv has finite mean and finite variance. Variance functions were invented to assist statistical analyses of agricultural experiments. They have since (under various names) found applications in basic and applied ecology (fisheries, forestry, wildlife), physics, meteorology, number theory, computer science, finance, generalized linear models, epidemiology, and demography, among other fields. Examples include Bartlett’s quadratic variance function and Taylor’s power law of fluctuation scaling, which Taylor did not discover. For example, the set of all primes that do not exceed *n* obeys variance = (1/3)mean^2 asymptotically as *n* becomes large. This talk will survey some uses and models of variance functions and recent generalizations of Taylor’s law to families of nonnegative rvs (stable laws) with infinite means and infinite variances.