QuickQuant和QuickVal的密度函数
Density functions for QuickQuant and QuickVal
摘要：
我们证明了，对于每一个$0 \leq t \leq 1$，由著名的算法QuickQuant使用的密钥比较的尺度归一化数的极限分布，以找到随机排序列表中的$t$第分位数具有Lipschitz连续密度函数$f_t$，该密度函数由$10$上界。此外，该密度$f_t(x)$对于每个$x > \min\{t, 1 - t\}$都是正的，并且在$t$中一致地在右尾享受超指数衰减。证明了生存函数$1 - F_t(x) = \int_x^{\infty}\!f_t(y)\,\mathrm{d}y$和密度函数$f_t(x)$都具有右尾渐近性$\exp [-x \ln x - x \ln \ln x + O(x)]$ .我们使用右尾渐近来约束QuickQuant使用的关键比较的标度归一化数的大偏差。
英文摘要：
We prove that, for every $0 \leq t \leq 1$, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the $t$th quantile in a randomly ordered list has a Lipschitz continuous density function $f_t$ that is bounded above by $10$. Furthermore, this density $f_t(x)$ is positive for every $x > \min\{t, 1 - t\}$ and, uniformly in $t$, enjoys superexponential decay in the right tail. We also prove that the survival function $1 - F_t(x) = \int_x^{\infty}\!f_t(y)\,\mathrm{d}y$ and the density function $f_t(x)$ both have the right tail asymptotics $\exp [-x \ln x - x \ln \ln x + O(x)]$. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.
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