The role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables
Sílvio M. Duarte Queirós
Unilever R&D Port Sunlight, Quarry Road East, Wirral, CH63 3JW UK
(10th December 2008)
In this manuscript we analyse the behaviour of the probability density function of the sum of deterministic variables generated from the triangle map of Casati-Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. When these properties are modified the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables increases the distance to Gaussian distribution increases. This behaviour goes against standard central limit theorem. By extrapolation of our finite size results we preview that in the limit of going to infinity the distribution has the same asymptotic decay as a Lorenztian (or a -Gaussian).
The central limit theory has been a subject of studied within the natural sciences for many generations. We might even state that CLT originated in 1713 with Bernoulli’s weak law of large numbers . After Bernoulli, de Moivre , Laplace , and Gauss, amongst others, made crucial contributions to the establishment of the Normal probability density function (PDF) as the stable distribution when one considers the sum of independent and identically distributed random variables with finite standard deviation. The stability of the Normal distribution, the central limit theorem (CLT), was just formally established by the Russian mathematician Lyapunov 188 years after Bernoulli . Afterwards, Lévy and Gnedenko [5, 6] generalised the CLT to account for independent and identically distributed random variables but with infinite standard deviations, followed by broader generalisations which include dependency between variables [7, 8]. With the advent of computation in the 1970s, chaos theory and non-linear phenomena achieved huge progress. It was then possible to verify the existence of a central limit theory for deterministic variables as well [9, 10]. More recently, central limit theory has been the focus of renewed interest within statistical mechanics mostly because of the endeavour to establish the optimising PDF of non-additive entropy  as the stable distribution for the sum of random or deterministic variables for some special kind of correlation , on the edge of chaos [13, 14], or even for systems in a metastable state .
In the sequel of this manuscript we communicate results on numerical
investigations of the distribution of deterministic variables which arise
from the sum of variables generated from the triangle map introduced by
Casati and Prosen . Our analysis is performed at two different
regimes: a first illustrative one, for which the system is both ergodic and mixing, and a second in which the system is weakly ergodic and apparently weakly mixing
2 The triangle map
where , and and are real parameters of the map. Function represents a modified definition of in the interval between and . This map has emerged from studies on the compatibility between linear dynamical instability and the exponential decay of Poincaré recurrences. Map (1) is parabolic and area preserving. For the Jacobian matrix we have, , and its trace, .
Concerning the relevance of parameters and , it is known that when both of the parameters are irrational numbers, the map is ergodic . Moreover, it attains the ergodicity property, i.e., averages over time equal averages over samples, very rapidly. For such a set of values the map is also mixing  in the sense that it has a continuous spectral density besides the property previously defined. Evaluating Poincaré recurrences, it has been found that the probability of an orbit to stay outside a specific subset of the torus for a time longer than behaves as , with being very close to the Lesbegue measure of that subset. This fact is in accordance with a completely stochastic dynamics. The exponential decay leads to a linear separation of close orbits which has been related to non-extensive statistical mechanics formalism via a generalised Pesin-like identity that bridges a -generalisation of Kolmogorov-Sinai entropy  and a -Lyapunov coefficient from the sensitivity to initial conditions . The entropic index of this map was found to be .
When and is irrational, the map is still ergodic , but is never mixing . For the case two situations might occur . If is a rational number, then the dynamics are pseudo-integrable and confined, whereas for irrational, the dynamics are found to be weakly ergodic, with the number of taken by a single orbit increasing as (). Furthermore, the ultra-slow apparent decay of the correlation function measured for this condition does not provide sufficient evidence of mixing. In recent years, some analytical attempts in order to characterise the map Eq. (1) have been made . Despite this, it has not been possible to prove or reject the mixing property for the Casati-Prosen map although strong indication of a non-mixing property prevails.
In this section, we present the numerical results of our study. Namely, we have considered variables and that are obtained from the addition of and variables of map (1),
that we analyse after detrending and rescaling, ( renders either or and is the standard deviation, ). In our survey we have neglected the case since it destroys the dependence of on and we have focussed on the following situations,
where the map is weakly ergodic. For each analysis, we have randomly placed a set of initial conditions (typically elements) within the torus and we have let the map run. The probability density functions and are then obtained from these initial conditions. Our numerical calculations have been performed using the Mathematica™ kernel which assures a symbolic computational procedure. Although analytical considerations in case I are in principle possible, we have skipped them because we have used case I as a benchmark of the peculiar behaviour of case II that we are going to present. It is also worth stating that, for case I, where analytical considerations have been made (or can be made), conditions of strong mixing and (semi)conjugation to a Bernoulli shift are mandatory . These conditions are not verified in our case II.
as goes to infinity, see Fig. 1. Moreover, as it occurs for the Lyapunov CLT, follows the scaling relation,
We have also verified a skew in our PDFs, for small , which are not visible in the PDFs of Ref. , but might be comprehended according to analytical work made on other types of maps [9, 13]. Explicitly, skewed distributions have analytically been found when studying the same problem using the dissipative, fully chaotic and strongly mixing map .
A completely different behaviour is found when case II is analysed. For this case, we have concentrated on , although for we have obtained the same qualitative results. Instead of distributions reminiscent of a Normal distribution, we have numerically observed probability distributions which are well described by,
with for every value of analysed, see Tab. 1 and Figs. 6 and 4. The method applied to determine has been the Meerschärt-Scheffler estimator (see Appendix) . The subsequent application of the Hill estimator  has given concordant results.
The upper bound of we have found () imposes that the standard deviation would diverge if the variable () was defined over the whole interval of real numbers. Since we are treating cases for which is finite, the support of the resulting PDFs is compact and defined between and for and . This obviously leads to a finite standard deviation, , for the cases we have studied. We have verified that does not follow the scaling relation Eq. (7), see Fig. 3. Instead, a power-law dependence can approximately be given with an exponent close to . In addition, we have observed that the shape of the distribution has strong similarity with the -stable Lévy distribution
()when plotted in a log-log scale, namely the emergence of an inflexion point before the straight line segment . We must emphasise that this similarity by no means implies a possible application of Lévy-Gnedenko generalisation of the CLT which states that, the probability density function of the sum of variables, each one associated with the same distribution Eq. (9) ( and ) converges in the limit as goes to infinity to a -stable Lévy distribution with . It is easy to verify that the results we report in this manuscript do not follow this theorem because; i) and variables are boxed up within intervals from to and hence their standard deviations are always finite, ii) due to the weak chaotic properties, variables and are not idenpendent at all instants .
Trying to infer about the scaling behaviour of with , a clear-cut power-law behaviour could not be found as it is visible in Fig. 5.
In the absence of clear power-law behaviour and using the fact that decreases as increases, we have tried to extrapolate a value of . To this end, inspired by finite-size scaling relations of critical behaviour , we have used the following ansatz,
From this, we have obtained (see Fig. 4) very close (within error margins) to the exponent of the Lorentzian distribution, .
4 Final remarks
In this manuscript we have presented a numerical experiment on the addition
of deterministic variables generated by a conservative map, the triangle map
of Casati and Prosen . The study has been performed in two
different regimes, case I and case II, by iterating the map from a set of
initial conditions which have uniformly been placed within interval . In case I, for which the map is ergodic and mixing, the outcoming stable
PDF is the Normal distribution for both and , in perfect
accordance with standard theory. In case II, for which the map is weakly
ergodic for sure and with apparently no mixing, we have obtained PDFs which
are well described by power-laws for large values of the variable. Moreover,
the parameter characterising the PDF is smaller than and it presents a
decreasing trend as the number of variables are augmented
SMDQ would like to thank Prof C. Tsallis for several discussions over several aspects of central limit theory, G. Ruiz for conversations about the application of CLT to deterministic variables, and T. Prosen for his comments on the properties of Eq. (1) and for having provided his work  before public disclosure. An acknowledgment is also addressed to Prof T. Cox for the thorough reading and commenting of this manuscript. The work herein presented has benefited from financial support by FCT/MCTES (Portuguese agency) and Marie Curie Fellowship Programme (European Union), and infrastructural support from PRONEX/CNPq (Brazilian agency).
The method introduced by Meerschaert and Scheffler  is based on the asymptotic limit of the sum of the variables of dataset under scrutiny. For heavy tail data these asymptotics depend only on the tail index of the probability density function, and not on the exact form of the distribution. Hence, if elements of a dataset are identically and (in)dependently distributed, and in addition its probability density function presents an asymptotic behaviour,
it can be proved (Theorem 1 in Ref. ) that,
where is the simple average and .
- email address: Silvio.Queiros@unilever.com, firstname.lastname@example.org
- By strongly ergodic we mean systems for which the exploration rate [( is the number of cells visited by a orbit until a discrete time and is the total number of cells ], averaged over several initial conditions, presents the same functional form as the exploration rate of a random model . Concomitantly, by strongly mixing we refer to systems whose correlation function, decays much faster than with . Weakly ergodic/mixing system do not follow each of corresponding the features .
- An exponent equal to corresponds to the lower bound for finite second order moment of distributions defined between and .
- can also work as a measure of time.
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