On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

Cuny, Christophe; Dedecker, Jérôme; Merlevède, Florence

doi:10.1016/j.spa.2017.07.011

Cited by 15 publications

(35 citation statements)

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“…In Section 2, following Korepanov [13], we represent the dynamical systems (1.1) and (1.5) as a function of the trajectories of a particular Markov chain; further, we introduce a meeting time related to the Markov chain and estimate its moments. In Section 4 we prove Theorem 1.3 for our new process (which is a function of the whole future trajectories of the Markov chain) by adapting the ideas of Berkes, Liu and Wu [2] and Cuny, Dedecker and Merlevède [3]. In Section 5 we generalize our results to the class of nonuniformly expanding dynamical systems and show the optimality of the rates.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 89%

“…The main difference between our situation and the one considered in [3] is that X n 's are functions of not only g n , but the whole future g n , g n+1 , . .…”

Section: Outlinementioning

confidence: 76%

“…The Markov chain (g n ) n≥0 behaves similarly to the Markov chain (W n ) n≥0 on the state space N, studied in [3,Sec The strategy used in [3] was to adapt the method of Berkes, Liu and Wu [2] for functions of iid r.v. 's to functions of Markov chains, in order to obtain sufficient conditions for the ASIP with rate o(n 1/p ) in terms of an L 1 -coupling coefficient.…”

Section: Outlinementioning

confidence: 99%

“…However, using the regularity of our observables, we shall see that it is possible to approximate X n by a measurable function of a finite number of coordinates. Then the proof in [2] can be adapted also to our situation, and the rate in the ASIP is, as in [3], related to the tail of the meeting time of the chain (g n ) n≥0 (see Section 3).…”

Section: Outlinementioning

confidence: 99%

“…To get the unconditional ASIP, we use the arguments given in [2, step 3.4]. So, as it is summarized in [3,Prop. 21], we infer that (4.11) will follow if one can prove that there exists r ∈ (2, ∞) such that ℓ≥k 0…”

Section: The Proofmentioning

confidence: 99%

See 4 more Smart Citations

Rates in almost sure invariance principle for slowly mixing dynamical systems

Cuny¹,

Dedecker²,

Korepanov³

et al. 2019

Ergod. Th. Dynam. Sys.

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We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau-Manneville intermittent maps, with Hölder continuous observables.Our rates have form o(n γ L(n)), where L(n) is a slowly varying function and γ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed O(n 1/4 ).To break the O(n 1/4 ) barrier, we represent the dynamics as a Young-towerlike Markov chain and adapt the methods of Berkes-Liu-Wu and Cuny-Dedecker-Merlevède on the Komlós-Major-Tusnády approximation for dependent processes. Theorem 1.3. Let γ ∈ (0, 1/2) and ϕ : [0, 1] → R be a Hölder continuous observable with ϕ dµ = 0. For the map (1.1), the random process S n (ϕ) satisfies the ASIP with variance c 2 given by (1.4) and rate o(n γ (log n) γ+ε ) for all ε > 0. For the map (1.5), the random process S n (ϕ) satisfies the ASIP with variance c 2 given by (1.4) and rate o(n γ ).The rates in Theorem 1.3 are optimal in the following sense:Proposition 1.4. Let f be the map (1.1). There exists a Hölder continuous observable ϕ with ϕ dµ = 0 such that lim sup n→∞ (n log n) −γ |S n (ϕ) − W n | > 0 for all Brownian motions (W t ) t≥0 defined on the same (possibly enlarged) probability space as (S n (ϕ)) n≥0 . Hence, one cannot take ε = 0 in Theorem 1.3.Remark 1.5. If c 2 = 0, the rate in the ASIP can be improved to O(1). Indeed, then it is well-known that ϕ is a coboundary in the sense that ϕ = u−u•f with some u : [0, 1] → R. By [7, Prop. 1.4.2], u is bounded, thus S n (ϕ) is bounded uniformly in n.Remark 1.6. It is possible to relax the assumption that ϕ is Hölder continuous. As a simple example, Theorem 1.3 holds if ϕ is Hölder on (0, 1/2) and on (1/2, 1), with a discontinuity at 1/2. See Subsection 4.3 for further extensions.Remark 1.7. Intermittent maps are prototypical examples of nonuniformly expanding dynamical systems, to which our results apply in a general setup, and so does the discussion of rates preceding Theorem 1.3. We focus on the maps (1.1) and (1.5) for simplicity only, and discuss the generalization in Section 5.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 89%

“…The main difference between our situation and the one considered in [3] is that X n 's are functions of not only g n , but the whole future g n , g n+1 , . .…”

Section: Outlinementioning

confidence: 76%