Let $F$ be a probability measure on $\mathbb{R}$ in the domain of attraction of a stable law with exponent $α\in (0, 1)$. We establish integral criteria on $F$ that significantly expand the probabilistic approach to Strong Renewal Theorems (SRTs). The criterion for $α\in (0,1/2]$ is much weaker than currently available ones and in some cases provides sufficient and necessary conditions for the SRT. The criterion for $α\in (1/2, 1)$ establishes the SRT in full generality and in a unified way, barring the Limit Local Theorems employed. As an application, for infinitely divisible $F$, an integral criterion on its Lévy measure is established for the SRT. As another application, for $F$ in the domain of attraction of a stable law without centering, an integral criterion on $F$ is established for the SRT for the ladder height process of a random walk with step distribution $F$.
Let $R_n$ be the number of distinct values of the $n$ simple samples from an infinite discrete distribution. In 1960 Bahadure proved $\displaystyle \lim_{n\to \infty} \frac{R_n}{\Enum R_n}=1$ in probability; Chen et al. proved the limit in the sense of almost sure convergence, along with other results. In this note we present results of CLT, MDP and LDP for $R_n$ under mild conditions.
We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to the domain of geometric partial attraction a semistable law with index $α\in (1/2,1]$. In the process we obtain local limit theorems for both finite and infinite mean, that is for the whole range $α\in (0,2)$. We also derive the asymptotics of the renewal function for $α\in (0,1]$.
Strong renewal theorems with infinite mean beyond local large deviations
设$F$是指数为$α\in(0,1/2]$的稳定律吸引区域中直线上的分布函数。通过推广Doney [Probab. Theory Related Fields 107（1997）451-465]。这是通过在$F$上引入条件来完成的，这些条件通常排除了$\mathbb{P}\{S_n\in(x,x+h]\}=O(n)\overline{F}(x)/x$类型的局部大偏差边界，因此明显弱于Doney（1997）中的有界性条件。并给出了结果在梯高过程和无穷可分分布中的应用。
Inspired by the double-debt problem in Japan where the mortgagor has to pay the remaining loan even if their house was destroyed by a catastrophic event, we model the lender's cash flow, by an exponential functional of a renewal-reward process. We propose an insurance add-on to the loan repayments and analyse the asymptotic behavior of the distribution of the first hitting time, which represents the probability of full repayment. We show that the finite-time probability of full loan repayment converges exponentially fast to the infinite-time one. In a few concrete scenarios, we calculate the e…
We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the ex…
When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the \emph{annealed} large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the \emph{quenched} LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an \emph{algebraic} tail. The rate function turns out to be a sum of two terms, one being the annea…
Infinite volume limits of polymer chains with periodic charges
本文的目的是双重的：-给一个明确的公式的自由能的一般类的聚合物链通过周期性的潜力与环境相互作用的一个基本的和自足的证明。这推广了[Bolthausen and Giacomin，AAP 2005]中的一个结果，其中公式是通过使用马尔可夫链的Donsker-Varadhan大偏差理论推导出来的。相反，我们利用更新理论的工具。- 确定系统的无限容量极限。特别是，在不同的状态中，我们会遇到瞬时、零循环和正循环过程（对应于轨迹的离域、临界和局部行为）。这是通过利用更新理论方法获得的系统的分拆函数的尖锐估计。系统无限体积极限的精确表征暴露了一个非唯一性问题。然而，我们将详细解释这种（最初）令人惊讶的现象是如何由于一阶相变的存在而引起的。
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $α\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $α$-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.
