Localization vs. delocalization of random discrete measures

Sequences of discrete measures $\mu^{(n)}$ with random atoms $\{\mu_i^{(n)}$, $i=1,2,\ldots\}$ such that $\sum_{i}\mu_i^{(n)}=1$ are considered. The notions of (complete) asymptotic localization vs. delocalization of such measures in the weak (mean or probability) and strong (with probability $1$) sense are proposed and analyzed, proceeding from the standpoint of the largest atoms' behavior as $n\to\infty$. In this framework, the class of measures with the atoms of the form $\mu_i^{(n)}=X_i/S_n$ ($i=1,\ldots,n$) is studied, where $X_1,X_2,\ldots$ is a sequence of positive, independent, identically distributed random variables (with a common distribution function $F$) and $S_n=X_1+\cdots +X_n$. If $\mathbb{E} [X_1] < \infty$, then the law of large numbers implies that $\mu^{(n)}$ is strongly delocalized. The case where $\mathbb{E} [X_1]=\infty$ is studied under the standard assumption that $F$ has a regularly varying upper tail (with exponent $0\le\alpha\le 1$). It is shown that for $\alpha < 1$, weak localization occurs. In the critical point $\alpha =1$, the weak delocalization is established. For $\alpha =0$, localization is strong unless the tail decay is “hardly slow”.