Renewal Theorem Research Articles

Recursive averages and the renewal theorem

Consider the sequence of numbers an defined by the iteration (1) for n ≥ k, where pr 1 ≤ r ≤ k) are non-negative numbers with and the starting values are given. So an is a weighted average of the previous k terms.

On the Convergence Rate in a Local Renewal Theorem for a Random Markov Walk

On the Convergence Rate in a Local Renewal Theorem for a Random Markov Walk

The Weight of a Ten-Pound Bag of Potatoes

Summary The weight of a ten-pound bag of potatoes is almost certainly not exactly ten pounds. Rather, it is a random variable with a minimum of ten pounds. We investigate the distribution of these weights in the context of a renewal theorem from the theory of stochastic processes and provide a straightforward demonstration of this theorem using basic tools from calculus.

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

AbstractWe study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in , , with respect to subordinate killed Brownian motions via ‐stable subordinators and establish the asymptotic behavior of the spectral heat content as for the full range of . Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when , where is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.

107.27 The discrete renewal theorem with bounded interevent times

107.27 The discrete renewal theorem with bounded interevent times

Average path length and Fermat distance in fractal networks composed of high-dimensional Sierpiński pyramids

Complex networks created by fractals have been associated with various applications such as social networks, modern communication, and fractal antennas. In this paper, we consider a series of scale-free networks based on the construction of high dimensional Sierpiński pyramids. Using self-similarity and elementary renewal theorem, we derive the asymptotic formulas of the average path length (APL) and the average Fermat distance (AFD) on our network sequence in turn. Our approaches are suitable for various networks modeled on self-similar fractals.It is known that the ratio of AFD to APL is between 3/2 and 2, which implies that AFD is also a key property to characterize small-world effect. Through an elaborate investigation of our networks, the limit of this ratio for our evolving networks is 3/2 and we analyze the result in the light of our interpretation. In fact, in some scale-free networks, the nodes with high degrees have more influence on the choice of Fermat points and thus Fermat distance. Our conclusions may lead to some interesting relations concerning the hyperbolicity, Laplacian spectrum and some multiparameter indices of networks.

Asymptotic formula on APL of fractal evolving networks generated by Durer Pentagon

Complex networks constructed by fractals have many applications in many fields, such as the data center networks and fractal antennas. In this paper, we consider a kind of evolving networks modeled on the classical fractal, Durer Pentagon, whose nodes are all the solid pentagons in the construction of Durer Pentagon up to stage t. In this network, two nodes are neighbors if and only if the intersection of their corresponding pentagons is a line segment. Using self-similarity and renewal theorem, we obtain the asymptotic formula on average path length (APL) of our evolving network.

Key renewal theorem and asymptotics of the renewal measure on Borel sets

Key renewal theorem and asymptotics of the renewal measure on Borel sets

Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn

For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the "power law P\'olya's urn" of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, Gonz\'alez Casanova, Kurt, and Span\`o.

Renewal theory for iterated perturbed random walks on a general branching process tree: Early generations

AbstractLet $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

Residue expansions and saddlepoint approximations in stochastic models using the analytic continuation of generating functions

Asymptotic residue expansions are proposed for inverting probability generating functions (PGFs) and approximating their associated mass and survival functions. The expansions are useful in the wide range of stochastic model applications in which a PGF admits poles in its analytic continuation. The error of such an expansion is a contour integral in the analytic continuation and saddlepoint approximations are developed for such errors using the method of steepest descents. These saddlepoint error estimates attain sufficient accuracy that they can be used to set the order of the expansion so it achieves a specified error. Numerical applications include a success run tutorial example, the discrete ruin model, the Pollaczek-Khintchine formula, and passage times for semi-Markov processes. The residue expansions apply more generally for inverting generating functions which arise in renewal theory and combinatorics and lead to a simple proof of the classic renewal theorem. They extend even further for determining Taylor coefficients of general meromorphic functions.

Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

Abstract We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity, then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistribution results. We apply these general results to obtain counting and equidistribution results for cusped Hitchin representations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.

Limit theorems for supercritical branching processes in random environment

The branching process in random environment {Zn}n≥0 is a population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with positive probability and grows exponentially fast on the nonextinction set. Using Fourier techniques we improve existing central limit theorem as well as we obtain Edgeworth expansions and the renewal theorem for the sequence {logZn}n≥0. The strategy is to compare logZn with partial sums of i.i.d. random variables in order to obtain precise estimates.

Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

AbstractAn iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of jth-generation individuals with birth times $\leq t$ , when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such generations form a subset of the set of intermediate generations.

Fourier decay, renewal theorem and spectral gaps for random walks on split semisimple Lie groups

Fourier decay, renewal theorem and spectral gaps for random walks on split semisimple Lie groups

Erratum: The remainder in the renewal theorem

Erratum: The remainder in the renewal theorem

Delayed Renewal Process with Uncertain Random Inter-Arrival Times

An uncertain random variable is a tool used to research indeterminacy quantities involving randomness and uncertainty. The concepts of an ’uncertain random process’ and an ’uncertain random renewal process’ have been proposed in order to model the evolution of an uncertain random phenomena. This paper designs a new uncertain random process, called the uncertain random delayed renewal process. It is a special type of uncertain random renewal process, in which the first arrival interval is different from the subsequent arrival interval. We discuss the chance distribution of the uncertain random delayed renewal process. Furthermore, an uncertain random delay renewal theorem is derived, and the chance distribution limit of long-term expected renewal rate of the uncertain random delay renewal system is proved. Then its average uncertain random delay renewal rate is obtained, and it is proved that it is convergent in the chance distribution. Finally, we provide several examples to illustrate the consistency with the existing conclusions.

One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

Trigonometric series and self-similar sets

Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $\mu$ on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. The rate of $\widehat{\mu}(\xi) \to 0$ is also shown to be logarithmic if $\log r_i / \log r_j$ is diophantine for some $i \neq j$. The proof is based on quantitative renewal theorems for stopping times of random walks on $\mathbb{R}$.

Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures * *The authors are supported in part by the National Natural Science Foundation of China, Grant 11771136, and Construct Program of the Key Discipline in Hunan Province. The first author is also supported in part by the Hunan Province Hundred Talents Program and a Faculty Research Scholarly

The spectral dimension of a fractal Laplacian encodes important geometric, analytic, and measure-theoretic information. Unlike standard Laplacians on Euclidean spaces or Riemannian manifolds, the spectral dimension of fractal Laplacians are often non-integral and difficult to compute. The computation is much harder in higher-dimensions. In this paper, we set up a framework for computing the spectral dimension of the Laplacians defined by a class of graph-directed self-similar measures on (d ⩾ 2) satisfying the graph open set condition. The main ingredients of this framework include a technique of Naimark and Solomyak and a vector-valued renewal theorem of Lau et al.