Reciprocal symmetry breaking in Pareto sampling

Abstract

Let $W_1,\ldots,W_N$ be a sample of $\mathrm{Pareto}(\alpha)$ random variables normalized by their sum, such that $\sum_i W_i=1$. The $W_i$ may represent the weights of valleys in a spin glass (if $0<\alpha<1$), or the frequency of different lineages (families) in a genealogy. This paper considers a population in which there are $N$ individuals reproducing with $\mathrm{Pareto}(\alpha)$ offspring-number distribution ($1<\alpha<2$). The probability of two randomly-chosen individuals being siblings, $Y_2=\sum_i W_i^2$, gives the sample mean of the normalized size of families, and its reciprocal gives the effective number of families (or reproducing lineages) in the population, $N_{\mathrm{e}}=1/Y_2$. The typical sample mean is very different from the average over all possible samples, i.e. $Y_2$ is not a self-averaging quantity. The typical $Y_2$ and its reciprocal do not vary with $N$ in opposite ways. Non-self-averaging effects are crucial in understanding genetic diversity in mass spawning species such as marine fishes.