Infinite measure renewal theorem and related results

Dolgopyat, Dmitry; Nándori, Péter

doi:10.1112/blms.12217

Cited by 17 publications

(10 citation statements)

References 32 publications

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“…We note that the main ingredient in most proofs is local limit theorem and its extensions, such as the Edgeworth expansion used in Section 4. This makes it plausible that similar results hold for other systems where the local limit theorem hold, including the systems described in [2,7,9,10,11,12,13]. Another natural research direction motivated by the present work is limit theorems for global observables.…”

Section: Discussionsupporting

confidence: 66%

Global observables for random walks: law of large numbers

Dolgopyat,

Lenci,

Nándori

2019

Preprint

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We consider the sums T N = N n=1 F (S n ) where S n is a random walk on Z d and F : Z d → R is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that T N always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on F , we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for quasiperiodic observables and for random walks in random scenery.

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Section: Discussionsupporting

confidence: 66%

Global observables for random walks: law of large numbers

Dolgopyat,

Lenci,

Nándori

2019

Preprint

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“…We recall that: a) [MT18] obtained mixing under mild abstract assumptions for, not necessarily Markov, suspension flows with regularly varying tails of roof functions of index in (1/2, 1]; b) [DN18b] obtained mixing for a class of Markov suspension flows with regular variation of index (0, 1).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Krickeberg mixing for Z extensions of Gibbs Markov semiflows

Terhesiu¹

2019

Preprint

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“…In this case, it suffices that τ 0 is Hölder continuous. Moreover, it follows from [18] that the mixing result (1.2) holds for all β ≤ 1. When c 1 is not an integer, f is not Markov and [18] does not apply, as far as we can tell, regardless of the value of β.…”

Section: Introductionmentioning

confidence: 86%

Renewal theorems and mixing for non Markov flows with infinite measure

Melbourne¹,

Terhesiu²

2020

Ann. Inst. H. Poincaré Probab. Statist.

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We obtain results on mixing for a large class of (not necessarily Markov) infinite measure semiflows and flows. Erickson proved, amongst other things, a strong renewal theorem in the corresponding i.i.d. setting. Using operator renewal theory, we extend Erickson's methods to the deterministic (i.e. noni.i.d.) continuous time setting and obtain results on mixing as a consequence.Our results apply to intermittent semiflows and flows of Pomeau-Manneville type (both Markov and nonMarkov), and to semiflows and flows over Collet-Eckmann maps with nonintegrable roof function.Here, ℓ : [0, ∞) → [0, ∞) is a measurable slowly varying function (so lim t→∞ ℓ(λt)/ℓ(t) = 1 for all λ > 0). Consider the suspension (Y τ , µ τ ) and suspension semiflow F t : Y τ → Y τ (the standard definitions are recalled in Section 3).The aim is to prove a mixing result of the formfor a suitable normalisation a t → ∞ and suitable classes of observables v, w :Under certain hypotheses, [13,37] obtained results on mixing and rates of mixing for such semiflows. The hypotheses were of two types: (i) assumptions on "renewal operators" associated to the transfer operator of F and the roof function τ , and (ii) Dolgopyat-type assumptions of the type used to obtain mixing rates for finite measure (semi)flows [17].As pointed out to us by Dima Dolgopyat, Péter Nándori and Doma Szász, mixing for indicator functions can be regarded as a local limit theorem and hence hypotheses of type (ii) should not be necessary.In this paper, we show that operator renewal-theoretic assumptions (i) are indeed sufficient for obtaining the mixing results in [13,37]. The abstract framework in [13] turns out again to be flexible enough to cover nonMarkov situations. Moreover, our main results extend to flows and we are able to treat large classes of observables v, w. (Conditions of type (i) alone are not sufficient for obtaining rates of mixing; the best results remain those in [13].)The analogous probabilistic results go back to Erickson [20] who obtained strong renewal theorems in an i.i.d. continuous time framework under the assumption β ∈ ( 1 2 , 1]. (In the discrete time setting, see [22] for the i.i.d. case and [35] for the deterministic case.) Our results on mixing when β ∈ ( 1 2 , 1] for semiflows (Corollary 3.1 and the extensions in Section 9) and for flows (Theorem 10.5), are proved by adapting Erickson's methods to the deterministic setting.For β ≤ 1 2 , additional hypotheses are needed on the tail of τ to obtain a strong renewal theorem (and hence mixing) even for discrete time; see [15,19,22] for i.i.d. results and [24] for deterministic results (see also [41] for higher order theory in both the i.i.d. and deterministic settings). For the continuous time case, Dolgopyat & Nándori [18] obtain strong renewal theorems for a class of Markov semiflows including the range β ≤ 1 2 (again under extra hypotheses on the tail µ(τ > t)), though our main examples seem beyond their framework. In the absence of additional tail hypotheses, [20] showed how to obtain a partial resul...

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Infinite measure renewal theorem and related results

Cited by 17 publications

References 32 publications

Global observables for random walks: law of large numbers

Global observables for random walks: law of large numbers

Krickeberg mixing for Z extensions of Gibbs Markov semiflows

Renewal theorems and mixing for non Markov flows with infinite measure

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