论文标题
限制续订数量的分布形式
Limit Forms of the Distribution of the Number of Renewals
论文作者
论文摘要
在这项工作中，探索了$ q_t（n）$的渐近属性，即在$ t $ $ t $中发生的续订数量（$ n $）的可能性。虽然很长一段时间（即$ t \ to \ infty $）的分布形式是非常众所周知的，并且与高斯中央限制定理或莱维稳定法律有关，但大量续订的替代限制，即$ n \ to \ to \ infty $。我们解决了大型$ n $的限制，并发现它达到了一种普遍形式，该形式仅取决于续订时间分布的分析属性。提供了$ q_t（n）$的显式公式，以及有限$ n $的校正以及融合通用渐近限制的必要条件。我们的结果表明，存在$ n/t $的较大偏差率函数，并在$ n/t \ to \ infty $限制中实现了通用线性增长（可达到对数校正）。该结果不论存在平均更新时间或幂律统计的存在。
In this work the asymptotic properties of $Q_t(N)$ ,the probability of the number of renewals ($N$), that occur during time $t$ are explored. While the forms of the distribution at very long times, i.e. $t\to\infty$, are very well known and are related to the Gaussian Central Limit Theorem or the Lévy stable laws, the alternative limit of large number of renewals, i.e. $N\to\infty$, is much less noted. We address this limit of large $N$ and find that it attains a universal form that solely depends on the analytic properties of the distribution of renewal times. Explicit formulas for $Q_t(N)$ are provided, together with corrections for finite $N$ and the necessary conditions for convergence to the universal asymptotic limit. Our results show that the Large Deviations rate function for $N/t$ exists and attains an universal linear growth (up to logarithmic corrections) in the $N/t\to\infty$ limit. This result holds irrespective of the existence of mean renewal time or presence of power-law statistics.
