AI 中文总结
研究N个独立同分布随机变量和的概率密度，开发高斯余项层次结构。对次指数密度，首个非平凡截断得简单有限N近似，含高斯卷积与减法，能捕获多区域，还再现渐近反常率函数并提供有限N率函数近似。
AI 中文摘要
我们为N个独立同分布随机变量之和的概率密度开发了一个精确的高斯余项层次结构，这些随机变量具有广泛的有限方差分布。该层次结构将高斯不动点贡献与保留原始单变量密度的残余部分分开。对于次指数密度，第一个非平凡截断产生一个简单的有限N近似，涉及与高斯背景的一次卷积和去除高斯重复计数的减法。数值结果表明，该近似在单个表达式中捕获了高斯中心、交叉区域和大跳跃尾部。相同的一阶近似再现了拉伸指数随机变量和的已知渐近反常率函数，并为相应的有限N率函数提供了精确近似。
英文摘要
We develop an exact Gaussian-remainder hierarchy for the probability density of the sum of $N$ independent, identically distributed random variables with broad, finite-variance distribution for the summands. The hierarchy separates the Gaussian fixed-point contribution from residual sectors that retain the original single-summand density. For subexponential densities, the first nontrivial truncation yields a simple finite-$N$ approximation that involves one convolution with a Gaussian background and a subtraction that removes Gaussian overcounting. This approximation captures the Gaussian center, the crossover region, and the big-jump tail within a single expression, as demonstrated numerically for stretched-exponential and finite-variance power-law examples. The same first-order approximation reproduces the known asymptotic anomalous rate function for sums of stretched-exponential random variables and also provides an accurate approximation to the corresponding finite-$N$ rate function.
Comments12 pages
