Séance Séminaire

Séminaire de Probabilités et Statistique

Taylor's Law of Fluctuation Scaling

A family of nonnegative random variables is said to obey Taylor's law when the variance of each random variable is proportional to some fixed power of the mean of that random variable: variance = a×mean^b with the same a>0 and b for all random variables in the family. E.g., in the family of exponential distributions, if the mean is \mu, then the variance is \mu² , so Taylor's law holds with a=1, b=2. The discrete-time Galton-Watson branching process, the continuous-time linear birth and death process, random multiplicative processes in independent or Markovian environments, and many other stochastic processes obey Taylor's law. Thousands of empirical illustrations of Taylor's law in many different sciences have been published. I will review some recent empirical and theoretical examples of Taylor's law. For example [1], in the limit as x->\infty, the mean M(x) and the variance V(x) of the prime numbers that do not exceed x asymptotically satisfy Taylor's law with a=1/3 and b=2: V(x)~(1/3){M(x)}². This and the exponential example show that the value of the exponent b of Taylor's law does not uniquely identify the process that generates the data. Also [2], nonnegative stable laws with no finite moments satisfy versions of Taylor's law. For example, in a sample of n independent observations from a nonnegative stable law with tail index \alpha where 0<\alpha<1, if M(n) is the sample mean and V(n) is the sample variance, then V(n)/{M(n)}b converges as n->\infty to a limiting random variable with finite mean and variance (that are known explicitly) if and only if b=(2-\alpha)/(1-\alpha), even though the population mean and all higher population moments of the stable law are infinite. Versions of Taylor's law hold for higher sample moments and the sample upper and lower semivariances of samples from nonnegative stable laws with no finite moments [3]. [1] Cohen, Joel E. 2016 Statistics of primes (and probably twin primes) satisfy Taylor's law from ecology. The American Statistician 70(4):399-404. DOI:10.1080/00031305.2016.1173591 [2] Brown, Mark, Cohen, Joel E. and de la Peña, Victor 2017 Taylor?s law, via ratios, for some distributions with infinite mean. Journal of Applied Probability 54(3):1-13. (September 2017) DOI:10.1017/jpr.2017.25 [3] Brown, Mark, Cohen, Joel E., Tang, Chuan-Fa, and Yam, Phillip. In preparation. Taylor's law of fluctuation scaling for the semivariances of heavy-tailed data.