发表机构
Google DeepMind; Faculty of Mathematics, University of Waterloo, Canada（谷歌DeepMind; 滑铁卢大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了可交换且符号不变随机变量部分和序列弱与强持续概率的最优上下界，证明两者均以 $n^{-1/2}$ 阶衰减，并补充了确定性情形及允许取零值的相关结果。
AI 中文摘要
设 $(X_1,\ldots,X_n)$ 为 $\mathbb{R}\setminus\{0\}^n$ 中既可交换又符号不变的随机变量。对每个 $k\in[n]$，令 $S_k=\sum_{i=1}^kX_i$。定义弱持续概率为 $\mathbb{P}(S_1,\ldots,S_n\geq0)$，强持续概率为 $\mathbb{P}(S_1,\ldots,S_n>0)$。根据已有结果，$\mathbb{P}(S_1,\ldots,S_n\geq0)$ 的最优下界和 $\mathbb{P}(S_1,\ldots,S_n>0)$ 的最优上界已知。我们通过确定 $\mathbb{P}(S_1,\ldots,S_n\geq0)$ 的最优上界和 $\mathbb{P}(S_1,\ldots,S_n>0)$ 的最优下界来完善这一图景，具体如下：\\[\frac{1}{2^n}\binom{n-1}{\lfloor (n-1)/2\rfloor}\leq\mathbb{P}(S_1,\ldots,S_n>0)\leq\frac{1}{4^n}\binom{2n}{n}\leq\mathbb{P}(S_1,\ldots,S_n\geq0)\leq\frac{1}{2^n}\binom{n}{\lfloor n/2\rfloor}.\\]特别地，这蕴含 $\mathbb{R}\setminus\{0\}^n$ 中每个可交换且符号不变的随机变量 $(X_1,\ldots,X_n)$ 的弱和强持续概率均为 $n^{-1/2}$ 阶。我们还获得了一些相关结果，一个在确定性情形下，另一个在随机变量允许取值为 0 的情形下。
英文摘要
Let $(X_1,\ldots,X_n)$ be a random variable in $(\mathbb{R}\setminus\{0\})^n$ that is both exchangeable and sign-invariant. For every $k\in[n]$, let $S_k=\sum_{i=1}^kX_i$. Define the weak persistence probability as $\mathbb{P}(S_1,\ldots,S_n\geq0)$, and the strong persistence probability as $\mathbb{P}(S_1,\ldots,S_n>0)$. From previous results, the optimal lower bound for $\mathbb{P}(S_1,\ldots,S_n\geq0)$ and the optimal upper bound for $\mathbb{P}(S_1,\ldots,S_n>0)$ are known. We complete the picture by determining the optimal upper bound for $\mathbb{P}(S_1,\ldots,S_n\geq0)$ and the optimal lower bound for $\mathbb{P}(S_1,\ldots,S_n>0)$ as follows. \[\frac{1}{2^n}\binom{n-1}{\lfloor (n-1)/2\rfloor}\leq\mathbb{P}(S_1,\ldots,S_n>0)\leq\frac{1}{4^n}\binom{2n}{n}\leq\mathbb{P}(S_1,\ldots,S_n\geq0)\leq\frac{1}{2^n}\binom{n}{\lfloor n/2\rfloor}.\] In particular, this implies that the weak and strong persistence probabilities of every exchangeable and sign-invariant random variable $(X_1,\ldots,X_n)$ in $(\mathbb{R}\setminus\{0\})^n$ are of the order $n^{-1/2}$. We also obtain some related results, one in the deterministic setting, and one when the random variables are allowed to take the value 0.
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