一个与光滑变换有关的min型随机不动点方程
A min-type stochastic fixed-point equation related to the smoothing transformation
摘要：
本文研究了随机不动点方程X \stackrel{d}{=} \inf_{i \geq 1: T_i > 0} X_i/T_i及其与加性方程$X \stackrel{d}{=} \sum_{i\ge 1}T_{i}X_{i}$的关系，并给出了光滑化变换.这里$\stackrel{d}{=}$表示分布相等，$T := (T_i)_{i \geq 1}$是一个给定的非负随机变量序列，$X, X_1, ...$是一个非负i.i.d.序列。独立于$T$的随机变量。我们提请注意的问题的存在性的非平凡的解决方案，特别是特殊的解决方案命名为$α$ -定期解决方案$(α>0)$。我们给出了一个完整的答案的问题时，$α$ -正规解决方案存在，并证明它们总是混合的威布尔分布或某些周期变量。我们也给出了这类不动点的完整刻画。分解方法，导致某些乘法鞅的研究和一个适当的变换后的路径更新方程是我们分析的关键工具。最后，我们给出了相应的可加性方程的不动点的结果。在某种程度上，这些结果已经由Iksanov早些时候获得。
英文摘要：
This paper is devoted to the study of the stochastic fixed-point equation X \stackrel{d}{=} \inf_{i \geq 1: T_i > 0} X_i/T_i and the connection with its additive counterpart $X \stackrel{d}{=} \sum_{i\ge 1}T_{i}X_{i}$ associated with the smoothing transformation. Here $\stackrel{d}{=}$ means equality in distribution, $T := (T_i)_{i \geq 1}$ is a given sequence of nonnegative random variables and $X, X_1, ...$ is a sequence of nonnegative i.i.d. random variables independent of $T$. We draw attention to the question of the existence of nontrivial solutions and, in particular, of special solutions named $α$-regular solutions $(α>0)$. We give a complete answer to the question of when $α$-regular solutions exist and prove that they are always mixtures of Weibull distributions or certain periodic variants. We also give a complete characterization of all fixed points of this kind. A disintegration method which leads to the study of certain multiplicative martingales and a pathwise renewal equation after a suitable transform are the key tools for our analysis. Finally, we provide corresponding results for the fixed points of the related additive equation mentioned above. To some extent, these results have been obtained earlier by Iksanov.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
