中心极限定理与多项式几何
Central limit theorems and the geometry of polynomials
摘要：
令$X \in \{0,\ldots,n \}$为随机变量，具有平均值$μ$和标准差$σ$，并令\[f_X(z) = \sum_{k} \mathbb{P}(X = k) z^k, \]为其概率母函数。Pemantle解释说，如果$σ$很大，而$f_X$没有接近$1\in \mathbb{C}$的根，那么$X$必须近似正常。我们完全解决了这个猜想在以下强定量形式，获得尖锐的界限。如果$δ= \min_ζ|ζ-1|$在$f_X$和$X^{\ast} := (X-μ)/σ$的复根$ζ$上，则为\[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{\log n}{δσ} \right) \]，其中$Z \sim \mathcal{N}(0,1)$为标准正态分布。这给出了Lebowitz，Pittel，Ruelle和Speer的结果的最佳可能版本。我们还表明，如果$f_X$没有小参数的根，那么$X$必须近似正常，再次在一个尖锐的定量形式：如果我们设置$δ= \min_ζ|\arg(ζ)|$然后\[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{1}{δσ} \right). \]使用这个结果，我们回答了一个问题的Ghosh，Liggett和Pemantle证明一个尖锐的多变量中心极限定理的随机变量与真实稳定的概率生成函数。
英文摘要：
Let $X \in \{0,\ldots,n \}$ be a random variable, with mean $μ$ and standard deviation $σ$ and let \[f_X(z) = \sum_{k} \mathbb{P}(X = k) z^k, \] be its probability generating function. Pemantle conjectured that if $σ$ is large and $f_X$ has no roots close to $1\in \mathbb{C}$ then $X$ must be approximately normal. We completely resolve this conjecture in the following strong quantitative form, obtaining sharp bounds. If $δ= \min_ζ|ζ-1|$ over the complex roots $ζ$ of $f_X$, and $X^{\ast} := (X-μ)/σ$, then \[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{\log n}{δσ} \right) \] where $Z \sim \mathcal{N}(0,1)$ is a standard normal. This gives the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. We also show that if $f_X$ has no roots with small argument, then $X$ must be approximately normal, again in a sharp quantitative form: if we set $δ= \min_ζ|\arg(ζ)|$ then \[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{1}{δσ} \right). \] Using this result, we answer a question of Ghosh, Liggett and Pemantle by proving a sharp multivariate central limit theorem for random variables with real-stable probability generating functions.
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