令$\{X_{i,j}:(i,j)\in\mathbb N^2\}$是随机变量$X$的独立副本的二维数组，并且令$\{N_n\}_{n\in\mathbb N}$是自然数序列，使得对于某个$c>0$，$\lim_{n\to\infty}e^{-cn}N_n=1$。我们的主要目标是独立随机乘积之和$$Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n}e^{X_{i,j}}.$$。结果表明，$Z_n$的极限性质，如$n\to\infty$，在两个临界点$c=c_1$和$c=c_2$经历相变。也就是说，如果$c>c_2$，那么$Z_n$满足中心极限定理与通常的正规化，而对于$c<c_2$，一个完全倾斜的$α$稳定的法律出现在极限。此外，$Z_n/\mathbb E Z_n$概率收敛于1当且仅当$c>c_1$。如果随机变量$X$是高斯的，我们恢复Bovier，Kurkova和Löwe的结果[REM和$p$ -自旋SK模型中自由能的波动。安。可能。30（2002），605-651]。
英文摘要：
Let $\{X_{i,j}:(i,j)\in\mathbb N^2\}$ be a two-dimensional array of independent copies of a random variable $X$, and let $\{N_n\}_{n\in\mathbb N}$ be a sequence of natural numbers such that $\lim_{n\to\infty}e^{-cn}N_n=1$ for some $c>0$. Our main object of interest is the sum of independent random products $$Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n}e^{X_{i,j}}.$$ It is shown that the limiting properties of $Z_n$, as $n\to\infty$, undergo phase transitions at two critical points $c=c_1$ and $c=c_2$. Namely, if $c>c_2$, then $Z_n$ satisfies the central limit theorem with the usual normalization, whereas for $c<c_2$, a totally skewed $α$-stable law appears in the limit. Further, $Z_n/\mathbb E Z_n$ converges in probability to 1 if and only if $c>c_1$. If the random variable $X$ is Gaussian, we recover the results of Bovier, Kurkova, and Löwe [Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30(2002), 605-651].
A study of the limiting behavior of delayed random sums under non-identical distributions setup
本文考虑S_{n+an} -Sn型延迟和，其中a_n是满足一定条件的可能正整数值随机变量，S_n是{G_1，G_2}中具有分布函数F_n的独立随机变量X_n的和.研究了时滞和的极限性态，证明了Chover型重对数律.这些结果推广了Vasudeva and Divanji（1992）和Chen（2008）的结果。
Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers
Achhruram Memorial College（阿克鲁拉姆纪念学院）；Ramakrishna Mission Vivekananda Educational and Research Institute（罗摩克里希那传教士维韦卡南达教育与研究院）；Kishori Sinha Mahila College（基绍里·辛哈女子学院）
