Sharp Error Terms for Return Time Statistics under Mixing Conditions

Abadi, Miguel; Vergne, Nicolas

doi:10.1007/s10959-008-0199-x

Cited by 33 publications

(78 citation statements)

References 13 publications

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“…We remark that in all the results mentioned so far there is a parameter ξ A n present, which converges in the limit, but allows a convenient perturbation of the asymptotic law H to improve the apparent convergence. The results we present here are of a similar form to [7], but are not restricted to balls and do not include this extra factor.…”

Section: A More Technical Introductionmentioning

confidence: 91%

“…However, the literature on dependent processes is much less extensive (for one case, see [36]). On the other hand, if we think of our stochastic process as coming from a Markov chain or, more generally, a dynamical system, then there is an equivalence between EVL and Hitting Time Statistics (HTS) (see [21]), which then yields a significant body of literature coming from that side on these error terms [25,1,2,7,5,31]. In this paper, taking inspiration from all these areas, we obtain sophisticated estimates on rates of convergence, where the dependence on time and 'length' (i.e., the distance from the maximum) scales is made explicit.…”

Section: Introductionmentioning

confidence: 99%

“…Further refinements were also made in [7]. A similar approach was used for α-mixing processes in [5], but the error terms were not as powerful, in particular, the time scale was not so nicely decoupled from length in the estimates.…”

Section: A More Technical Introductionmentioning

confidence: 99%

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Speed of convergence for laws of rare events and escape rates

Freitas

Todd

2015

Stochastic Processes and their Applications

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We obtain error terms on the rate of convergence to Extreme Value Laws for a general class of weakly dependent stochastic processes. The dependence of the error terms on the `time' and `length' scales is very explicit. Specialising to data derived from a class of dynamical systems we find even more detailed error terms, one application of which is to consider escape rates through small holes in these systems

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Section: A More Technical Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Speed of convergence for laws of rare events and escape rates

Freitas

Todd

2015

Stochastic Processes and their Applications

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“…Namely, they consider the cases when ζ is a periodic point and obtain the existence of an EI less than 1, although they did not state in these terms because, at the time, the connection with EVL was not yet established. In fact, Galves and Schmitt [41] introduced a short correction factor λ in order to get exponential HTS, that was then studied later in great detail by Abadi et al [32,[42][43][44][45][46], and which, in case of being convergent, can actually be seen as the EI itself.…”

Section: Existence Of Extreme Value Laws For Chaotic Systemsmentioning

confidence: 99%

Extremal behaviour of chaotic dynamics

Freitas

2013

Dynamical Systems

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We present a review of recent results regarding the existence of extreme value laws for stochastic processes arising from dynamical systems. We gather all the conditions on the dependence structure of stationary stochastic processes in order to obtain both the distributional limit for partial maxima and the convergence of point processes of rare events. We also discuss the existence of clustering which can be detected by an extremal index less than 1 and relate it with the occurrence of rare events around periodic points. We also present the connection between the existence of extreme value laws for certain dynamically defined stationary stochastic processes and the existence of hitting times statistics (or return times statistics). Finally, we make a complete description of the extremal behaviour of expanding and piecewise expanding systems by giving a dichotomy regarding the types of extreme value laws that apply. Namely, we show that around periodic points we have an extremal index less than 1 and at very other point we have an extremal index equal to 1. IntroductionThe purpose of this work is to compile a series of recent results regarding the theory of extreme events for dynamical systems. Extreme means that we are interested in the occurrence of rare but possibly catastrophic events. This motivates its applications to situations where risk assessment is rather crucial.Instead of focusing on means and average behaviour of the system, often ruled by central limit theorems, in this approach, we are interested on the largest (or smallest) observations of sample data in order to understand the extremal behaviour of the system by obtaining corresponding distributional limits.The starting point of our analysis is a stationary stochastic process arising from a chaotic dynamical system simply by evaluating an observable function along the orbits of the system, as explained in Section 3. Then the extremal behaviour is analysed by studying the distributional limit of the partial maximum of such stochastic processes or by investigating the limit of point processes counting the number of exceedances of high thresholds during some time interval.In the context of dynamical systems, the study of extreme value laws (EVL) is a quite recent topic. It appeared first in the pioneering paper of Collet [1], which has been an

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“…This law can also be well approximated by an exponential law with parameter being the measure of the string. However, when the string overlaps itself the limiting law is a convex combination of a Dirac measure at the origin and an exponential law [2], [6], [8]. As in the case of the hitting time, the parameter must be corrected by the factor given above.…”

Section: Introductionmentioning

confidence: 99%

Rényi Entropies and Large Deviations for the First Match Function

Abadi

Cardeño

2015

IEEE Trans. Inform. Theory

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We define the first match function T n : C n → {1, . . . , n} where C is a finite alphabet. For two copies of x n 1 ∈ C n , this function gives the minimum number of steps one has to slide one copy of x n 1 to get a match with the other one. For ergodic positive entropy processes, Saussol and coauthors proved the almost sure convergence of T n /n. We compute the large deviation properties of this function. We prove that this limit is related to the Rényi entropy function, which is also proved to exist. Our results hold under a condition easy to check which defines a large class of processes. We provide some examples.

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Sharp Error Terms for Return Time Statistics under Mixing Conditions

Cited by 33 publications

References 13 publications

Speed of convergence for laws of rare events and escape rates

Speed of convergence for laws of rare events and escape rates

Extremal behaviour of chaotic dynamics

Rényi Entropies and Large Deviations for the First Match Function

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