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On discrete stochastic processes with long-lasting time dependence in the variance S´ılvio M. Duarte Queirós 1 Unilever R&D Port Sunlight, Quarry Road East, Wirral, CH63 3JW UK (19th September 2008) 9 0 0 Abstract 2 n a In this manuscript, we analytically and numerically study statistical properties of an het- J eroskedastic process based on the celebrated ARCH generator of random variables whose 2 2 variance is defined by a memory of q -exponencial, form (ex = ex). Specifically, we m qm=1 inspect the self-correlation function of squared random variables as well as the kurtosis. ] n In addition, by numerical procedures, we infer the stationary probability density func- a - tion of both of the heteroskedastic random variables and the variance, the multiscaling a t properties, the first-passage times distribution, and the dependence degree. Finally, we a d introduce an asymmetric variance version of the model that enables us to reproduce the . so-called leverage effect in financial markets. s c i s y 1 Introduction h p [ Many of the so-called complex systems are characterised by having time series with a 2 peculiar feature: although the quantity under measurement presents an autocorrelation v function at noise level for all time lags, when the autocorrelation of the magnitudes is 7 1 appraised, a slow and asymptotic power-law decay is found. This occurs, e.g., with (log) 6 price fluctuations of several securities traded in financial markets [1], temperature fluctua- 2 . tions [2], neuromuscular activation signals [3] or even fluctuations in presidential approval 6 0 ratings [4] amongst many others [5]. Moreover, most of these time series are also char- 8 acterised by probability density functions with asymptotic power-law decay and a profile 0 suggestive of intermittency that is identified by regions of quasi-laminarity interrupted : v by spikes. In this perspective, this type of time series might be seen as a succession of i X measurements with a time-dependent standard deviation. Mathematically, this type of r a stochastic process is defined as heteroskedastic, in opposition to the class of processes with constant standard deviation that is defined as homoskedastic. With the primary goal of reproducing and forecasting inflation time series, it was introduced in 1982 the autoregressive conditional heteroskedasticity process (ARCH) [6]. The ARCH process rapidly has come to be a landmark in econometrics giving raise to several generalisations and widespread applications not only in Economics and Finance but in several other fields as well. In the sequel of this article, we introduce further insight into a variation of the ARCH processstudiedinRef.[7]whichisabletoreproducethepropertieswehavereferredtohere 1Previousaddress: CentroBrasileirodePesquisasF´ısicas,RuaDr. XavierSigaud150,22290-180,Rio de Janeiro-RJ,Brazil email address: [email protected],[email protected] 1 above. Our considerations are made both on analytical and numerical grounds. Although the primary goal of this manuscript is an extensive description of the model following the lines of Ref. [7], some assessment of its capability of reproducing the same features of SP500 daily log fluctuations spanning the 3rd January 1950 to the 28th February 2007 is made. In this context, we also introduce a slight modification on the model which turns it able to reproduce the leverage effect when the model is applied to surrogate price fluctuations time series. The manuscript is organised as follows: after introducing the ARCH processes and present some general properties, we make known in Sec. 2 some analytical calculations on the autocorrelation function of the model herein analysed and the correlation between variables and squared standard deviation for the extension as well as the expressions for the kurtosis. In Sec. 3, we introduce results from the numerical analysis about the probability density functions of the stochastic variable, z , and its t squared instantaneous standard deviation, σ2; The dependence degree between zt and t t z2 ; The distribution of first passage times of z2; and the multiscaling properties. In t+τ t Sec. 4, we establish an asymmetric variation of the model which allows the reproduction of the so-called leverage effect. Final considerations are addressed to Sec. 5. 2 The symmetric variance model We start defining an autoregressive conditional heteroskedastic (ARCH) time series as a discrete stochastic process, z , t z = σ ω , (1) t t t where ω is an independent and identically distributed random variable with mean equal t to zero and second-order moment equal to one, i.e., ω = 0 and ω2 = 1. Usually, ω is h ti h ti associated with a Gaussian distribution, but other distributions of ω have been presented to mainly describe price fluctuations [8]. In his benchmark article of Ref. [6], Engle suggested a dynamics for σ2 establishing it as a linear function of past squared values of t z , t s σ2 = a+ b z2 , (a,b 0). (2) t i t i i ≥ − i=1 X In financial practise, namely price fluctuation modelling, the case s = 1 (b b) is, by far, 1 ≡ the most studied and applied of all ARCH-like processes. It can be easily verified, even for all s, that, although zt zt′ δtt′, correlation zt zt′ is not proportional to δtt′. h i ∼ h| | | |i As a matter of fact, for s = 1, it has been proved that, z2 z2 decays as an exponential h t t′i law with a characteristic time τ lnb −1, which does not reproduce empirical evidences. ≡ | | In addition, the introduction of a large value for parameter s bears implementation problems [10]. Expressly, large values of s soar the complexity of finding the appropriate set of parameters b for the problem under study as it corresponds to the evaluation of a large i { } number of fitting parameters. Aiming to solve the imperfectness of the original ARCH process, the GARCH(s,r) process was introduced [11], with Eq. (2) being replaced by, s r σ2 = a+ b z2 + c σ2 (a,b ,c 0), (3) t i t i i t i i i ≥ − − i=1 i=1 X X 2 Nonetheless, eventhisprocess,presentsaexponentialdecayforhzt2 zt2′i,withτ ≡ |ln(b+c)|−1 for GARCH(1,1), though condition, b + c < 1, guarantees that GARCH(1,1) corresponds exactly to an infinite-order ARCH process [12]. Despite the fluctuation of the instantaneous volatility, the ARCH(1) process is actu- ally stationary with a stationary variance, given by, a σ2 = σ2 = , (b > 1), (4) 1 b − (cid:10) (cid:11) ( ... represents averages over samplebs and ... averages over time). Moreover, it presents h i a stationary probability density function (PDF), P (z), with larger kurtosis than the kurtosis of distribution P(ω). The fourth-order moment is, c 1+b z4 = a2 ω4 (1 b)(1 b2 σ4 ). − − h i (cid:10) (cid:11) (cid:10) (cid:11) This kurtosis excess is precisely the outcome of time-dependence of σ . Correspondingly, t when b = 0, the process reduces to generating a signal with the same PDF of ω, but with a standard variation √a. In the remaining of this article we consider a ARCH(1) process where an effective immediate past return, z˜ , is assumed in the evaluation of σ2 [7]. Explicitly, Eq. (2) is t 1 t − replaced by σ2 = a+bz˜2 , (a,b 0), (5) t t−1 ≥ where the effective past return is calculated according to t 1 − z˜2 = (i t+1) z2, (6) t−1 K − i Xi=t0 where 1 t ′ (t) = exp , (t 0,T > 0,q < 2) (7) K ′ (t) qm T ′ ≤ m Zqm ′ (cid:20) (cid:21) with 1 exp [x] = ex [1+(1 q) x]1−q , (q ), (8) q q ≡ − + ∈ R (t) 0 exp i ([x] = max 0,x 2),knownintheliteratureasq-exponential [13]. Zqm ′ ≡ i= t′ qm T + { } This prososal c−an be enclosed in the fractionally integrated class of heteroskedastic pro- P (cid:2) (cid:3) cess (FIARCH). Although it is similar to other proposals [14], it has a simpler structure which permits some analytical considerations without introducing any underperformance when used for mimicry purposes. For q = , we obtain the regular ARCH(1) and for −∞ q = 1, we have (t) with an exponential form since exp [x] = ex [15]. Although it has a K ′ 1 non-normalisable kernel, let us refer that the value q = corresponds to the situation m ∞ that all past values of z have the same weight, t 1 (t) = . (9) ′ K t t +1 ′ 0 − 2This condition is known in the literature as Tsallis cut off at x= (1 q)−1. ± − 3 In this case, memory effects are the strongest possible, i.e., every single element of that past has the same degree of influence on σ2 making it constant after a few steps. Because t of this, in this case, P(z) is the same as noise ω, as shown in [7]. Similar heuristic arguments are the base for the Gaussian nature of the “Elephant random walk”[16]. Assuming stationarity in the process some calculations can be made3. Namely, it is provable that the average value of σ2 yields, a σ2 = σ2 = , (b > 1), (10) 1 b − (cid:10) (cid:11) and the covariance, ztzt′ correspondbs to h i ztzt′ = σtσt′ ωtωt′ , (11) h i h ih i which, due to the uncorrelated nature of ω, gives hztzt′i = 0 for every t 6= t′ and hzt2i = σ2 . In addition, we can verify that all odd moments of z are equal to zero. Concerning t h i the fourth-order moment, z4 , we have h ti z4 = σ2σ2ω2ω2 = σ2 2 ω4 , (12) t t t t t t t D E (cid:10) (cid:11) (cid:10) (cid:11) (cid:2) (cid:3) (cid:10) (cid:11) which by expansion yields, t 1 z4 = a2 ω4 +2ab − (i t+1) z2 ω4 + h ti h ti K − h iih ti i=t0 P t 1 b2 − [ (i t+1)]2 z4 ω4 + (13) K − h iih ti i=t0 P t 1 t 1 2b2 − − (i t+1) (j t+1) z2z2 ω4 . K − K − i j h ti i=t0j=i+1 P P (cid:10) (cid:11) If z and z are assumed as strictly independent, then z2z2 = z2 z2 = z2 2. i j i j h ii j h t i Assuming stationarity we have, (cid:10) (cid:11) (cid:10) (cid:11) z4 = a2 ω4 +2ab z2 ω4 + h ti h ti h tih ti t 1 b2 z4 ω4 − [ (i t+1)]2+ h tih ti K − (14) i=t0 P t 1 t 1 2b2 z2 2 ω4 − − (i t+1) (j t+1), h t i h ti K − K − i=t0j=i+1 P P or a2 +2ab z2 +2b2 z2 2 z4 = h ti h t i Q1 ω4 , (15) t I 1 b2 ω4 t t 2 − h iQ (cid:10) (cid:11) (cid:10) (cid:11) 3Thishasbeennumericallyanalysedbycomputingthezt2 self-correlationfunctionfordifferentwaiting times 4 t 1 t 1 t 1 with = − − (i t+1) (j t+1) and = − [ (i t+1)]2. On the 1 2 Q K − K − Q K − i=t0j=i+1 i=t0 other hand, we hPavePthe other limiting case, z2z2 = z4 . EqPuation (13) is then written i j h t i as a2 +(cid:10)2ab z(cid:11)2 z4 = h ti ω4 . (16) t C 1 b2 ω4 (2 + ) t t 1 2 − h i Q Q The labelling as upper bo(cid:10)und(cid:11)for z4 and lower bound f(cid:10)or (cid:11)z4 comes as follows; The h tiI h tiC introduction of non-Gaussianity in heteroskedastic processes comes from the fluctuations in the variance (or in z2), when the variables are strongly attached between them, there t is a small level of fluctuation in σ2 and eventually it becomes constant. With σ being t t a constant, or approximately that, there is not introduction of a significant level of non- Gaussianity measured from z4 , hence z4 > z4 (a = 0). h ti h tiI h tiC 6 For an accurate description of z4 , which lies between the two limiting expressions, h ti we must compute correlations z2z2 . That is obtained averaging, h t t′i t 1 z2z2 = a2ω2ω2 +ab − (i t+1) z2ω2ω2+ t t′ t t′ K − i t t′ i=t0 P t′ 1 ab − (j t +1) z2ω2ω2+ (17) K − ′ j t t′ j=t0 P t 1 t′ 1 b2 − − (i t+1) (j t +1) z2z2ω2ω2. K − K − ′ i j t t′ i=t0j=t0 P P Defining τ t t, the last term of rhs of Eq. (17), hereon labelled as , is responsible ′ ≡ − C for the dependence of z2z2 with τ. It can be written as h t t′i t 1 t τ 1 = b2ω2ω2 − − − (i t+1) (j t+1+τ) z2z2 C t t′ K − K − i j i=t0 j=t0 P P t τ 1 − − (i t+1) (i t+1+τ) z4+ (18) ∼ K − K − i i=t0 P t τ 1 t 1 − − − (i t+1+τ) (j t+1) z2z2. K − K − i j i=t0 j=t τ+1 − P P We shall now consider a continuous approximation where the summations are changed by integrals, t 1 t − ... ...dx, → Xi=t0 Z0 and t 1, so that the following relations are obtained in the limit t . Computing ≫ → ∞ , the term in z4 has the coefficient, hCi i t τ C1(τ) ∼ 0− expqm[x−t]expqm[x−t+τ] dx (19) R 1 t→=∞ [τ(qm2−−1q)m]1−qm F2,1 q2m−−qm1, qm1−1;2+ qm1−1; (1−q1m)τ , h i 5 where F [a,b;c;x] is the generalised hypergeometric function. Regarding the term in 2,1 z2z2, the first approximation is obtained considering z2z2 = z2 z2 = z2 2. Its i j i j h ii j h ti coefficient is then given by (cid:10) (cid:11) (cid:10) (cid:11) t t τ C2(τ) ∼ t τ 0− expqm[y −t]expqm[x−t+τ] dxdy − (20) R R exp [ λτ], t∼ qc − →∞ with 1 q = , (21) c 2 q m − and λ = q 1. A simple inspection shows that (τ) decays much slower than (τ), c− C2 C1 hence the asymptotic form of z2z2 is dominated by (τ) as it is illustrated in the inset h t t′i C2 of Fig. 1. In Fig. 1, we bring face to face Eq. (20) and the autocorrelation function of z2 from numerical simulation using the parameters applied to reproduce SP500 returns t previously determined, namely a,b = b ,q = q = 0.5,0.99635,1.6875 . SP m SP { } { } From all these equations we are able to conjecture expressions which relate parameters a,b,q with the form of the distribution for the case where z4 is finite. In this way, { m} h ti we can use the ansatz that the distribution of this dynamical model is associated with a q-Gaussian (or Student-t) distribution 4, p(z) = exp z2 , (q < 3), (22) q A −B with = σ¯2(3 q) −1, where, (cid:2) (cid:3) B q − (cid:2) (cid:3) σ¯2 z2[p(z)]qdz/ [p(z)]qdz, q ≡ Z Z is the q-generalised second order moment [13], and is the normalisation constant. This A assumption is based on the same type of arguments used in [27] and whose accuracy we verify later on (see Sec. 3). For q < 5/3, σ¯2 relates to the usual variance according to q σ¯2(3 q) = σ¯2(5 3q). q − − From Eqs. (13), (17), (19), and (20) we can write, hzt4i = a2hωt4i+2ab hzt2ihωt4i+b2hzt4ihωt4i (23−qqmm)2 − (23) +b2 ω4 a2 ω2 2 +2ab z2 ω2 2 +2 . h ti h ti h ti h ti K h i where represents terms like, K t t exp [x t]exp [x+τ t][ (τ)+ (τ)] dτ dx, (24) qm − qm − C1 C2 Z0 Z0 4Aq-Gaussian,withq >1,correspondstoaStudent-twithmdegreesoffreedomwithq = 3+m where 1+m m is taken as a real positive number. 6 whichcorrespondstoaquitecomplexintegrationoverxofhypergeometricfunctionspecial ˜ ˜ functions like F a˜,b;c˜;x and the Appell hypergeometric function [17] where a˜, b, and 2,1 c˜represent generahl values.i Taking into attention that for a q-Gaussian with q < 7 its fourth moment is, 5 3q 5 z4 = 3 σ¯2 2 − , (25) 5q 7 − (cid:10) (cid:11) (cid:0) (cid:1) we can obtain approximate relations between the parameters of the model and the parameters of the distribution. This is achieved when we equalise Eqs. (23) and (25), remember- ing the expression of the variance, Eq. (10), and the form of the autocorrelation function of z2. This procedure is obviously important in parameter estimation. Therefore, from t the decay of the z2 autocorrelation function we can determine the value of q , and a and t m b from the equalisation we have just referred to together with Eq. (10). 3 Numerical considerations 3.1 Stationary probability density functions We firstly recover the results previously presented for the adjustment of P (z) with q- Gaussians. As mentioned in [7] the adjustment is striking (diagrams of q as a function of bandq arepresentedinFigure1ofthatreference). Usingthemethodofχ2 minimisation, m we have obtained for the same cases previously studied5 average values of χ2 = 1.1 10 6 − × (per degree of freedom) and R2 = 0.99990. In Fig. 2, we present an example for which it is possible to assent the accuracy of the fitting not only in the tails, but in the peak of the PDF as well. We have performed further analysis using the cumulative distribution function (CDF) and the Kolmogorov-Smirnov Distance, D , KS D = max H(z) H (z) , (26) KS 0 | − | where H(z) is the empirical CDF obtained from numerical evaluation of the model and H (z) is the testing probability density function, 0 z H (z) = exp x2 dx. (27) 0 A q −B Z −∞ (cid:2) (cid:3) The average D value obtained for the same cases plotted in Figure 1 of the prior work KS is equal to 4.25 10 3. Such values allow us to rely on the null hypothesis [19], − × P (z) = p(z) = exp z2 . (28) A q −B Based on the acceptance of the null hypothesis(cid:2)(28) w(cid:3)e are able to introduce some insight into the distribution of σ2, p (σ2). Firstly, we carry out the following change of σ variables, z˘ = ln z2 t t σ˘ = ln σ2 , (29)  t t ω˘ = ln ω2  t t 5Comparing with prior studies we have increased the runs by a factor of 10. 7 so that Eq. (1) turns into, z˘ = σ˘ +ω˘ . (30) t t t In probability space, regarding that σ˘ and ω˘ are independent, we have, t t ∞ p(z˘) = P (σ˘) P (z˘ σ˘) dσ˘. (31) σ˘ ω˘ − Z −∞ We can now apply the convolution theorem. Being p(z˘), the probability of z˘, according t t to such a theorem, p(z˘) = 1 Pˇ (k)Pˇ (k) , (32) t − σ˘ ω˘ F where (cid:2) (cid:3) 1 Pˇ (k) = f (x)exp[ikx] dx [f (x)], (33) x √2π ≡ F Z and 1[f (k)] = 1 f (k)exp[ ikx] dx = f (x) is the inverse Fourier Transform. F− x √2π x − Since we respectively know and postulate the form of p(ω) and p(z), we can write down, R p(ω˘) = 1 exp ω˘ eω˘ , √2π 2 − 2 (34) h i p(z˘) = exp ez˘ exp z˘ , A q −B 2 ( = (σ¯ = 1)), yielding the respective Fou(cid:2)rier Tr(cid:3)ansfor(cid:2)m(cid:3)s [20], B B Pˇ (k) = 2ik−21Γ 1 +ik , ω˘ π 2 (cid:2) (cid:3) Pˇz˘(k) = √12πA (−1)−Q(Bq −B)1−1q−QB 1 q,Q, q2−1q + (35) B−B − n h i Γ 1 +ik F˜ 1 , 1 +ik, 3 +ik, q . 2 2,1 q 1 2 2 B −B − h io (cid:2) (cid:3) where Q = 1 +1+ik, B[...] is the Beta function, and F˜ [...] is the regularised hyper- 1 q 2 2,1 geometric fu−nction [17]. Applying Eq. (35) in Eq. (32) we can compute the distribution of σ˘ (easily related to p (σ)), σ Pˇ (k) P (σ˘) = 1 z˘ , (36) σ˘ F− Pˇ (k) (cid:20) ω˘ (cid:21) ˜ From a laborious and tricky calculation, using properties of F [...] (see [18] and related 2,1 properties), it can be verified P (σ˘) cooresponds to, σ˘ Γ[θ ik] Pˇ (k) = − . (37) σ˘ ik (2β) Γ[c] Therefore, σ2 follows an inverse Gamma distribution, 1 1 pσ σ2 = (2θ)c Γ[c] σ2 −1−cexp −2θσ2 , (38) (cid:20) (cid:21) (cid:0) (cid:1) (cid:0) (cid:1) 8 where c = 3 q and θ = q 1 . This result attests the validity of the superstatistical 2q−2 σ¯2(5−3q) approach to t−he problem of he−teroskedasticity. It is worth mentioning that superstatistics [21] represents the long-term statistics in systems with fluctuations in some characteristic intensive parameter of the problem like the dissipation rate in Lagragian turbulent fluids [22] or the standard deviation like in the subject matter of heteroskedasticity. For the values of q and b , with have obtained random variables z associated with a SP SP t (q = 1.465)-Gaussian. This yields c = 1.648... and θ = 0.770..., which have been SP SP applied in Fig. 3 to fit p (σ2) obtained by numerical procedures. In that plot, it is visible σ that numerical and analytical curves are in proximity. 3.2 Dependence degree The degree that the elements of a time series are tied-in is not completely expressed by the correlation function in the majority of the cases. In fact, regarding its intimate relation with the covariance, the correlation function is only a measure of linear dependences. Aiming to assess non-linear dependences, information measures have been widely applied [23]. In our case, we use a non-extensive generalisation of Kullback-Leibler information measure [24, 25]6, 1 q p′(z2,z2 ) − t t+τ 1 p(z2,z2 ) − I p z2,z2 (cid:20) t t+τ (cid:21) , (39) q ≡ − t t+τ 1 q t − X (cid:0) (cid:1) where p z2,z2 = p(z2)p z2 = [p(z2)]2 (assuming stationarity), which has been ′ t t+τ t t+τ able to provide a set of interesting results with respect to dependence problems [26]. The (cid:0) (cid:1) (cid:0) (cid:1) quantification of the dependence degree is made through a value, qop, which corresponds to the inflexion point of the normalised version of I , q I q R , (40) q ≡ Imax q where Imax is the value of I when variables z(t) and z(t+τ) present a biunivocal de- q q pendence (see full expression in Ref. [25]). Forinfinitesignalsitcanbeshownthat,whenthevariablesarecompletelyindependent qop = , whereas qop = 0 when variables are one-to-one dependent. For finite systems, ∞ there is a noise level, qop, which is achieved after a finite time lag τ. Typical curves of R n q are depicted in Fig. 5 for q = q and b = b . m SP SP In what follows, we present results obtained from numerical evaluation of R for differ- q ent values of τ. As expected, dependence relies on the balance between the extension of memory, which isgiven byq andtheweight ofeffective pastvalue, z˜ , onσ . Firstly, let m t 1 t − us compare cases q = q ,b = b and q = 1.25,b = b , as an example of what m SP SP m SP { } { } happens when we fix b as a constant (see Fig. 4). Dependence is obviously long-lasting in the former case, in the sense that it takes longer to attain qop, but for small values of n 6In the limit q 1 the Kullback-Leibler mutual information definition is recovered. → 9 τ, the latter has presented higher levels of dependence. This has to do that (t) is nor- ′ K malised and that implies the intersection of the curves for different values of q at some m value of t. Alternatively, when q decreases, the recent values of z have more influence ′ m t on z˜ than past values. When the value of q is kept constant, we have verified that t 1 m small−er values of b lead to a faster approach to noise value qop. In a previous work on n GARCH(1,1) [27], we have verified that variables approximately associated with the samedistributionpresentthesamelevelofdependenceindependentlyofthepair(b,c)cho- sen. In this case, recurring to cases q = 1.375,b = 0.75 and q = 1.625,b = 0.875 , m m { } { } we have noticed that the curves present very close values for small lags, but they fall apart forτ > 10, revealing a moreintricate relationbetween q, q , andbthaninGARCH(1,1). m Additionally, comparing dependence and correlation, we have verified that the decay is faster for the latter. Specifically, taking into account noise values of qop and C (z2), it is τ t verifiable that qop takes longer to achieve qop than C (z2) takes to reach C (z2). n τ t n t 3.3 First-Passage Times First-passagestudiesinstochasticprocessesareofconsiderableinterest. Notonlyfromthe scientific point of view [28] (it is useful in the approximate calculation of the lifetimes of the problems/systems) as well as from a practical perspective, since they can be applied to quantify the extent of reliability of forecasting procedures, e.g., in meteorology or finance [1, 29, 30]. In what it is next to come, we have analysed the probability of z2 S = [a,b) and z2 S . We have divided z2 domain into five different intervals. t ∈ i t+T ∈ i t Explicitly: S : z2 1; • 1 t ≤ S : 1 < z2 2; • 2 t ≤ S : 2 < z2 5; • 3 t ≤ S : 5 < z2 10; • 4 t ≤ S : z2 > 10. • 5 t Analysing the probability density functions we have verified that the simplest expression which enables a numerical description of first-passage inverse cumulative probability distribution, (t), is a linear composition of a asymptotic power-law (or a ν-exponential) D with a stretched exponential, t 1−1ν t φ (t) = ǫ 1+(ν 1) +(1 ǫ)exp . (41) D − T − − T (cid:20) 1(cid:21) " (cid:18) 2(cid:19) # CurvesofsomeanalysedcasesarepresentedinFig.6andfittingparametersinTab.1. The caseswepresentare: I- q = q ,b = 0.5 ,II- q = 1.25,b = b ,III- q = 1.25,b = 0.5 , m SP m SP m { } { } { } IV- q = 1.5,b = 0.875 . m { } From the Fig. 6 we have verified that, excepting region S , all curves of (t) exhibit 1 D a decay closely exponential (ν = 1). For region S , as we increase the non-Gaussianity of 1 10