Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources

Abstract

Motivated by entropy estimation from chaotic time series, we provide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a n-cylinder fluctuates in the same way as the inverse measure of this n-cylinder at 'small scales', but in a different way at 'large scales'. In particular, the Rényi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more difficult to handle theoretically. The results we obtain for return times, though less complete, improve the available ones.

Generalized dimensions, large deviations and the distribution of rare events

“…They quantify the rate at which the probability of observing "non typical" values diminishes when the radius goes to zero. Our result, Proposition 2 in Section 3, parallels a previous investigation [21] where the target set in phase space is a dynamical cylinder, rather than a ball. Rigorous theory is presented for the case of conformal repellers 1 but we believe that similar results hold in more general settings.…”

“…In this paper we have explored the relations between the spectrum of generalized dimensions D q and the recurrence properties of the dynamics. In fact, the former determines the large deviations of dynamical quantities such as return times [29] and hitting times: [21] and Proposition 2 herein. The statistics of hitting times ruled by Proposition 1 also opens the way to new techniques to estimate generalized dimensions via recurrence properties.…”

“…A related result will be derived herein, by considering hitting times [21], rather than return times: hitting times are related to targeting small regions of phase space, starting from a different, arbitrary point. Proposition 1 in Section 3 shows that dimensions indexed by q ≤ 2 (the meaning of this index, which is related to the order of moments, will be explained in the next section) can be computed in this way.…”

On Match Lengths, Zero Entropy, and Large Deviations—With Application to Sliding Window Lempel–Ziv Algorithm

“…Motivated by Kontoyiannis' results and the satisfaction of large deviation property for Shannon-McMillan-Breiman Theorem, it is natural to ask under what conditions the asymptotic recurrence times relation satisfies the large deviation property. Chazottes and Ugalde [9] in 2005 established partial large deviations results on recurrence times for Gibbsian sources. In this paper, we display a class of processes for which large deviation property holds completely for recurrence times.…”

“…One can consider a general setting where the process is assumed to be stationary and totally ergodic with zero entropy and the asymptotics of the first return time follow the law given by Eq. (9). Here, we can choose L o given by Eq.…”