Strong Renewal Theorem Research Articles

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.

Strong Renewal Theorem and Local Limit Theorem in the Absence of Regular Variation

We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to the domain of geometric partial attraction of a semistable law with index \(\alpha \in (1/2,1]\). In the process we obtain local limit theorems for both finite and infinite mean, that is, for the whole range \(\alpha \in (0,2)\). We also derive the asymptotics of the renewal function for \(\alpha \in (0,1]\).

On the ratio of current age to total life for null recurrent renewal processes

A number of open problems associated with determining the limit distribution of the ratio of current age to total life for a null recurrent renewal process (i.e. where inter-arrival times have infinite mean) are solved. In particular, when the survival function for the inter-arrival times satisfies F̄(t)∼t−αL(t) as t→∞ with L slowly varying and 0≤α≤1, we prove that the limit distribution corresponds to that of U1∕α, where U is uniformly distributed on (0,1), with the limit distribution taken to be degenerate at 0 when α=0. By using direct methods instead of appealing to strong renewal theorems, we are able to prove this result without regard to whether the inter-arrival time distribution is latticed or not, and without extraneous constraints on the renewal function.

Renewal theorems and mixing for non Markov flows with infinite measure

Nous obtenons des résultats de mélange pour une large classe de flots et de semi-flots préservant une mesure de masse infinie (et qui ne sont pas necessairement markoviens). Erickson a prouvé, entre autres choses, un théorème de renouvellement fort dans le contexte de variables aleatoires indépendantes et identiquement distribuées. En utilisant la théorie des opérateurs de renouvellement, nous étendons les méthodes d’Erickson au cas du temps continu déterministe (et donc on i.i.d.) et en déduisons des résultats sur le mélange. Nos résultats s’appliquent a des semi-flots et flots intermittents de type Pomeau–Manneville (à la fois de type markoviens ou de type non-markoviens) ainsi qu’à des suspensions au-dessus de transformations de Collet–Eckmann pour lesquelles la fonction toit est non-intégrable.

Local large deviations and the strong renewal theorem

We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha$. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long standing problem, which dates back to the 1962 paper of Garsia and Lamperti [Comm. Math. Helv.] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Pacific J. Math.] for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790

Strong renewal theorems and local large deviations for multivariate random walks and renewals

We study a random walk $\mathbf{S} _n$ on $\mathbb{Z} ^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha } =(\alpha _1,\ldots ,\alpha _d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, i.e. a sharp asymptotic of the Green function $G(\mathbf{0} ,\mathbf{x} )$ as $\|\mathbf{x} \|\to +\infty $, along the “favorite direction or scaling”: (i) if $\sum _{i=1}^d \alpha _i^{-1} < 2$ (reminiscent of Garsia-Lamperti’s condition when $d=1$ [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0} ,\mathbf{x} )$, sharpening estimates when $\mathbf{x} $ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha _i\equiv \alpha $, in the favorite scaling, and has even left aside the case $\alpha \in [1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.

On a Multivariate Strong Renewal Theorem

This paper takes the so-called probabilistic approach to the strong renewal theorem (SRT) for multivariate distributions in the domain of attraction of a stable law. A version of the SRT is obtained that allows any kind of lattice–nonlattice composition of a distribution. A general bound is derived to control the so-called small-n contribution, which arises from random walk paths that have a relatively small number of steps but make large cumulative moves. The asymptotic negligibility of the small-n contribution is essential to the SRT. Applications of the SRT are given, including some that provide a unified treatment to known results but with substantially weaker assumptions.

Strong renewal theorems with infinite mean beyond local large deviations

Let $F$ be a distribution function on the line in the domain of attraction of a stable law with exponent $\alpha\in(0,1/2]$. We establish the strong renewal theorem for a random walk $S_1,S_2,\ldots$ with step distribution $F$, by extending the large deviations approach in Doney [Probab. Theory Related Fileds 107 (1997) 451-465]. This is done by introducing conditions on $F$ that in general rule out local large deviations bounds of the type $\mathbb{P}\{S_n\in(x,x+h]\}=O(n)\overline{F}(x)/x$, hence are significantly weaker than the boundedness condition in Doney (1997). We also give applications of the results on ladder height processes and infinitely divisible distributions.

Second Order Expansions for the Moments of Minimum Point of an Unbalanced Two-sided Normal Random Walk

In this paper, the second order expansions for the first two moments of the minimum point of an unbalanced two-sided normal random walk are obtained when the drift parameters approach zero. The basic technique is the uniform strong renewal theorem in the exponential family. The comparison with numerical values shows that the approximations are very accurate. It is shown, particularly, that the first moment is significantly different from its continuous Brownian motion analog while the second moments are the same in the first order. The results can be used to study properties of the maximum likelihood estimator for the change point.

A strong renewal theorem for generalized renewal functions in the infinite mean case

LetF(x) be a nonarithmetic c.d.f. on (0, ∞) such that 1 —F(x)=x−αL(x), whereL(x) is slowly varying and 0≤α≤1. Leta(x) be regularly varying with exponent β≥1. A strong renewal theorem (of Blackwell type) for generalized renewal functions of the form\(G(t) \equiv \sum\limits_{n = 0}^\infty {a(n) F^n (t)} \) is proved here, thus extending the recent work of Embrechts, Maejima and Omey [1] and that of Erickson [4].

A note on a strong renewal theorem

AbstractLet {Sn, n ≥ 1} be a sequence of partial sums of independent and identically distributed non‐negative random variables with a common distribution function F. Let F belong to the domain of attraction of a stable law with exponent α, 0 &lt; α &lt; 1. Suppose H(t) = ϵ N(t), t ⩾ 0, where N(t) = max(n : Sn ≥ t). Under some additional assumptions on F, the difference between H(t) and its asymptotic value as t → ∞ is estimated.

Strong Renewal Theorems with Infinite Mean

Let F be a nonarithmetic probability distribution on $(0,\infty )$ and suppose $1 - F(t)$ is regularly varying at $\infty$ with exponent $\alpha ,0 < \alpha \leqq 1$. Let $U(t) = \Sigma {F^{{n^ \ast }}}(t)$ be the renewal function. In this paper we first derive various asymptotic expressions for the quantity $U(t + h) - U(t)$ as $t \to \infty ,h > 0$ fixed. Next we derive asymptotic relations for the convolution ${U^ \ast }z(t),t \to \infty$, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function $m(t) = \smallint _0^t[1 - F(x)]dx$, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case $\alpha = 1$ to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.

Strong renewal theorems with infinite mean

Let F be a nonarithmetic probability distribution on ( 0 , ∞ ) (0,\infty ) and suppose 1 − F ( t ) 1 - F(t) is regularly varying at ∞ \infty with exponent α , 0 &gt; α ≦ 1 \alpha ,0 &gt; \alpha \leqq 1 . Let U ( t ) = Σ F n ∗ ( t ) U(t) = \Sigma {F^{{n^ \ast }}}(t) be the renewal function. In this paper we first derive various asymptotic expressions for the quantity U ( t + h ) − U ( t ) U(t + h) - U(t) as t → ∞ , h &gt; 0 t \to \infty ,h &gt; 0 fixed. Next we derive asymptotic relations for the convolution U ∗ z ( t ) , t → ∞ {U^ \ast }z(t),t \to \infty , for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function m ( t ) = ∫ 0 t [ 1 − F ( x ) ] d x m(t) = \smallint _0^t[1 - F(x)]dx , t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case α = 1 \alpha = 1 to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.