0} and {bn 2 R} such that lim P ✓ MnY bn  x◆ = G(x), with G non-degenerate, n!1 an then (Gnedenko, 1943) G(x)⌘EV⇠(x)=⇢ exp (1+⇠x) 1/⇠ ,1+⇠x>0, if ⇠6=0, exp( exp( x)), x 2 R, if ⇠ = 0, is the well-known extreme value (EV) cumulative distribution function (CDF), being ⇠ the extreme value index (EVI). Let us next think on the stationary sequence {Xn}n 1. If the conditions which enable us to guarantee the existence of an extremal index (EI), denoted by ✓, held (Leadbetter et al., 1983), then lim P✓MnX bn x◆=EV✓⇠(x), 0✓1. n!1 an Indeed, we shall exclude the ‘slight pathological’ case ✓ = 0. Remark 2.1. Note that the EV⇠ CDF is max-stable, and consequently, EV✓⇠(x)=EV⇠✓x ✓◆, ✓=✓⇠ 1, ✓=✓⇠. ✓ ⇠ To better understand the intuitive meaning of the EI, let us think on the point process of exceedances over high thresholds. For any ⌧ > 0, let un = un(⌧), n 1, be a level such that F(un)=1 ⌧/n+o(1/n), asn!1, (2.1) a so-called normalized level, i.e. a level such that n(1 F(un))!⌧>0, or equivalently, a level such that F n (un ) ! exp( ⌧ ), as n ! 1. Note that it is possible to choose such a level for any CDF such that (1 F (x ))/(1 F (x)) ! 1, as x ! 1, which will be assumed throughout. Then, with IA denoting the indicator function of A, Y Xn Sn = I{Yj>un} j=1 ]]>un} j=1 converges, as n ! 1, towards a compound Poisson. Then P(MnY  un) = P(SnY = 0) ! e ⌧, whereas P(MnX  un) = P(SnX = 0) ! e ✓⌧, as n!1. This means that the intensity ⌧ of the Poisson limiting exceedances, in the IID case, becomes ✓⌧, i.e. the point process limit for the time normalized upcross- ings of high levels is also a Poisson point process but with intensity ✓⌧. Under independence (or even adequate quite weak dependence), this point process converges to a homogeneous Poisson process, as n ! 1, but when there is a slightly stronger local dependence, clusters of exceedances may occur and the limiting process of exceedances may be a compound Poisson process. Indeed, for a large class of weakly dependent processes, an upcrossing is generally followed by a cluster of exceedances and therefore the clusters may be roughly identified by the occurrence of upcrossings. Indeed, Leadbetter and Nandagopalan (1989) proved that the EI can then also be defined as the reciprocal of the ‘mean time of duration of extreme events’, and it is directly related to the exceedances of high levels. We have ✓ = 1 = lim P(X2 un|X1 >un) limiting mean size of clusters n!1 = lim P(X1  un|X2 > un), n!1 where un is a sequence of values such that (2.1) holds. Then, P(Yn:n  un) = Fn(un) ! e ⌧ and P(Xn:n  un) ! e ✓⌧. n!1 n!1 Remark 2.2. The limiting distribution of normalized maximum values of both {Xn} and {Yn} is thus of the same type, but there exists a ‘shrinkage’ of maximum values. This leads to ‘clusters of exceedances of high levels’ with a mean size greater than 1. To illustrate these facts, we next provide in Figure 1, sample paths associated with {2.1} an IID Fr´echet sequence, {Yi}i 1, from F(x) = exp{ x ↵}, x 0, ↵ = 2 (⇠ = 1/2 = 0.5, ✓ = 1), {2.2} a stationary 2-dependent sequence also with Fr´echet margins and ↵ = 2, defined as Xi = 2 1/↵ max (Yi, Yi+1) , Yi given in {2.1}, i 1, for which ✓ = 0.5, and ]]> 0, 0< <1.Then, Xn =Vn2 issuchthat Xn =AnXn 1 +Bn, n 1, X0 0, (3.2) An = Wn2, Bn = Wn2, {(An, Bn), n 1} IID R2+-valued random pairs. The stochastic di↵erence equation in (3.2) was introduced by Kesten (1973) and further studied in Vervaat (1979), where several examples are provided, like for instance the stock of material checked at regular time intervals, with An the intrinsic decay or increase of the stock and Bn the quantity added or taken ]]> 0. Let us consider the process in (3.2). If there is a positive real ↵ such that EA↵n=1, EA↵nln+An<1, 0u,Xj+1u] j=1 [Xj uu] j=1 I[Xj >u] To have consistency, the high level u must be such that n(1 F(un)) = cn⌧ = ⌧n, ⌧n ! 1 and ⌧n/n ! 0 (Nandagopalan, 1990). But whenever dealing with EVI-estimation we often consider an intermediate sequence kn, i.e. a sequence k=kn suchthatkn !1,butk/n!0,asn!1.Suchakn-sequencehas been replaced, in an EI-estimation, by the sequence ⌧n = cn⌧ with cn ! 1 as n ! 1. To make the semi-parametric EI-estimation closer to the most common semi-parametric EVI-estimation, and denoting by {Xi:n}1in the sample of ]]> 0, and G denote the Multivariate Extreme Value distribution of X. The tail dependence function (Huang, 1992; Schmidt and Stadtmu¨ller, 2006) of G is defined by `(x ,...,x ) = logG(x 1,...,x 1), (x ,...,x ) 2 [0,1)d. 1d1d1d It is a convex function, homogeneous of order one and satisfies _d j=1 xj  `(x1,...,xd)  Xd j=1 xj, with the lower bound corresponding to X with totally dependent margins and the upper bound to X with independent margins. The tail dependence function `(x1,...,xd) gives us informaWtion about the probability of occurrence of extreme events for the maximum dj=1 F(Xj). In fact, we have (Schmidt and Stadtmu¨ller, 2006; Ferreira and Ferreira, 2012a, 2012b) `(x1,...,xd)= lim ⇣ tlogP⇣F(X1)1 x1,...,F(Xd)1 xd⌘⌘ t!1⇣ xt x⌘t = lim tP F(X1)>1 1 _···_F(Xd)>1 d . t!1 t t It holds that `(x,...,x) = x`(1,...,1) and, for i(S) = 1 if i 2 S and i(S) = 0 i f i 2/ S , `( 1(S),..., d(S)) = ✏(XS), (2.1) where ✏(XS) is the extremal coe cient of the subvector XS of X with indices in S (Tiago de Oliveira, 1962-63; Smith, 1990). It takes values in the interval [1,|S|], where |S| is the number of elementsVin S, with ✏(XS) = 1 when XS has the minimum copula CXS (u1,...,ud)S = Q uj and ✏(XS) = |S| when XS j2S has the product copula CXS (u1,...,ud)S = j2S uj. W Let I = {I1,...,Ip} be a partition of D = {1,...,d}, M(Ij) = i2Ij Xi and XIj the subvector of X with indices in Ij. For given j 2 (0,1), j = 1,...,p, we will summarize the extremal de- pendence among the weighted subvectors 1 XIj , j = 1, . . . , p, through the j coe cient R(X, , I) defined as follows. Definition 2.1. Let X be a vector of unit Fr´echet random variables and Mul- tivariateExtremeValuedistribution.Foreach =( 1,..., p)2(0,1)p and ]]> xj} MAX-MIN DEPENDENCE COEFFICIENTS 19 p ✏(XIj ) ✏(XIj ) j=1 |T|+1 j2T XX X X ˆ (c) R(X,1,I)= Proof. If X has MEV distribution then it is associated (Marshall and Olkin, 1983) and then the variables M (Ij ), j = 1, . . . , p, are also associated (Esary et al., 1967). For these associated variables it holds that p ( 1) 1+X✏(XIj). 1 + ✏(XIj ) ;6=T ✓{1,...,p} j2T j=1 P (M(Ij) > xj) P and 0 1 P (M(Ij) > xj) j=1 j=1 j=1 ^p \p Yp P (M(Ij)  xj) P @ {M(Ij)  xj}A P (M(Ij)  xj). j=1 j=1 j=1 By taking Mˆ(Ij) = Wi2Ij Xˆi and M¯(Ij) = Wi2Ij X¯i, j = 1,...,p, we can rewrite the above inequalities as 0\p 1 0\p 1 0\p 1 P@ {M¯(Ij)>xj}A P@ {M(Ij)>xj}A P@ {Mˆ(Ij)>xj}A j=1 j=1 j=1 and 0\p 1 0\p 1 0\p 1 P@ {M¯(Ij)xj}A P@ {M(Ij)xj}A P@ {Mˆ(Ij)xj}A. j=1 j=1 j=1 This concordance order implies (Shaked and Shanthikumar, 2007) that 0 0 _p E @ f @ 1 1 0 0 _p M¯ ( I j ) A A  E @ f @ 1 1 0 0 _p 1 1 M ( I j ) A A  E @ f @ Mˆ ( I j ) A A , j=1 and 0 0^p 11 0 0^p 11 0 0^p 11 E @ f @ M¯ ( I j ) A A E @ f @ M ( I j ) A A E @ f @ Mˆ ( I j ) A A j=1 j=1 j=1 for all non-decreasing functions f. The result in (a) follows by taking f = F and replacing Xi by Xi , i 2 Ij, for each j = 1,...,p. j j=1 j=1 ]]>⇢b 2 (h),h=1,2,..., t |X|X where ⇢bX denotes the empirical autocorrelation function of X. This fact is now known as Taylor e↵ect. The corresponding theoretical relation is called the Taylor property, and assuming that the functions ⇢|X| and ⇢X2 are positive, is expressed by the Accepted: 23 March 2015. 2010 Mathematics Subject Classification. 62M10. Key words and phrases. Autoregressive and Bilinear models, kurtosis, Taylor property. This work was partially supported by the Centre for Mathematics of the Univer- sity of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Part- nership Agreement PT2020. 25 ]]>⇢X2 (h), h2N, with ⇢X denoting the autocorrelation function of X. The presence of this property in a certain class of models for time series can help in the selection of a more appropriate model to describe the dynamics of the series of interest. However, this study requires the knowledge of moments of X of order higher than 2 which makes such analysis theoretically di cult. Since the Taylor e↵ect was initially detected in financial time series, it is nat- ural that the first theoretical studies have focused on models used in the analysis of these series, specifically the conditionally heteroskedastic models, as can be found in He and Ter¨asvirta (1999), Gon¸calves, Leite and Mendes-Lopes (2009), Haas (2009) and Leite (2013), whose studies involve the class of Generalized Threshold Autoregressive Conditionally Heteroskedastic (GTARCH) models. Interestingly, these studies have shown that the presence of the Taylor prop- erty appears to have a stronger relationship with the weight of the tails of the process than with the characteristic of heteroskedasticity. This finding leads us to question the presence of the Taylor property in models without conditional heteroskedasticity but still suitable for financial time series analysis. The modeling of non-negative time series has been widely regarded in the literature (Zang and Tong, 2001) and in Tsay and Chan (2007) the interest of non-negative ARMA processes in the context of stochastic modeling of fi- nancial time series is referred. Additionally, the recent finding that the Taylor e↵ect is clearly present in some non-negative physical time series (Gonc¸alves et al., 2014) also leads us to question the presence of the property in bilinear models, which reveal themselves useful in the analysis of time series recorded, for instance, in seismology, hydrology and astronomy. So, this paper focuses on non-negative stochastic models and, given the complexity of the calculations, we limit our study to the first-order AR and bilinear models. Returning to Gonc¸alves, Martins and Mendes-Lopes (2014, 2015) we begin by recalling the fundamental results on the occurrence of the Taylor property in non-negative autoregressive and bilinear models of order 1. We point out that the property is analyzed for all lag h in the case of linear models. On the other hand, in the case of the bilinear model our study is limited to h = 1, due to the di culty associated with the calculations of the autocorrelations. The presence of the Taylor property is then assured in the linear case when the error process is symmetrically or right-skewed distributed and new examples are developed considering autoregressive and bilinear models with error processes presenting di↵erent values of kurtosis, being notorious the importance of this parameter on the presence of the Taylor property in the model. ]]>1 2E(Xt)μ3,X. V ( X t2 ) Let us note that, as KX 1, we may conclude that if μ3,X 0 then the Taylor property is present in the process X, or equivalently, if the genera- tor process is symmetric or right-skewed distributed, taking into account that μ3,X = μ3,"/(1 3). ]]> 3. Also, when tends to 1, KX tends to 3, independently of the value of K". Examples We evaluate now the presence of the Taylor property in model (2.1) con- sidering generator processes with non-negative distributions and weight tails significantly di↵erent. We take into account the necessary and su cient condi- tion of the presence of the Taylor property, written in the form T ( )=1 2m Cov Xt,Xt2 >0, (2.2) L 1 V ( X t2 ) where L denotes the marginal law of the generator process, ". Generator process with platykurtic distribution We consider the process X defined in (2.1) in cases where "t is distributed according to a platykurtic Beta probability law, Beta(↵, ✓), whose density has the form f1(x) = (↵+✓) x↵ 1(1 x)✓ 1 I]0,1[(x), ↵ > 0, ✓ > 0. More precisely, (↵) (✓) KX( )=1+ 2 +1+ 2K". we analyze the cases ✓ = ↵ (symmetrical distribution), ✓ = 2↵ (right-skewed distribution) and ✓ = ↵ (left-skewed distribution). Concerning the values of 2 thekurtosis,wehaveK" =31+2↵ when✓=↵,K" =32+3↵ when✓= ↵ and 3+2↵ 4+3↵ 2 K" =31+3↵ when✓=2↵.Inthecases✓=↵and✓=2↵,Corollary2.3 2+3↵ allows us to conclude that the Taylor property is present. In both cases, the functions KX( ) and TL( ) are also functions of ↵ and the presence of the Taylor property becomes, in general, weaker when KX increases. Comparing now the cases ✓ = ↵ and ✓ = 2↵, it is easy to verify that, for each ↵, the kurtosis is greater for the right-skewed distribution. In Figure 1 we present the graphs of the functions KX( ) related to the cases Beta(↵,↵) and Beta(↵,2↵), as well as the graphs of TBeta(↵,↵)( ) and TBeta(↵,2↵)( ), 0< <1, 0<↵<5. ]]> 0, ✓ > 0, that is, with density f2(x) = ↵✓ e ↵xx✓ 1I]0,+1[ (x), whose kurtosis coe cient is K" = (✓) 3+6.So,K ( )= 6 2 +1 2 3+6 alsodependson✓(butnoton↵)asit ✓ X 1+ 2 1+ 2 ✓ happens with T (✓,↵)( ). The Taylor property is always present in model (2.1) as the distribution (✓, ↵) is right-skewed. We note that when ✓ increases then T (✓,↵) ( ) approaches zero, that is, the autocorrelation functions in study are closer as ✓ increases. This result is in agreement with the evolution of the kurtosis, taking into account that now the process is more leptokurtic when ✓ decreases. Analogous conclusions are obtained when we consider "t following a Pareto distribution with parameters ↵and✓,Par(↵,✓),withdensityf (x)= ✓↵✓ I (x),↵>0,✓>4(inorder 3 x✓+1 ]↵,+1[ to assure the existence of m4). The kurtosis is K" = 3 + 6(✓3+✓2 6✓ 2) which ✓(✓ 3)(✓ 4) is a decreasing function of ✓ that goes to 9 when ✓ tends to infinity, and to infinity when ✓ tends to 4. So, the Pareto distribution is leptokurtic, no matter the value of ✓. These conclusions are illustrated in Figure 3, which shows the graphs of the functions KX ( ), related to the Gamma and Pareto distributions, and those of the functions T (✓,↵) ( ) and TPar(↵,✓) ( ), 0 < < 1, 4 < ✓ < 10. 4 800 K 10 Θ 0 0 4 1 T 10 Θ 0 0 ΦΦ 11 Figure 3. Graphs for K = KX( ) (on the left) and T = TL( ) (on the right) in the cases Par(↵,✓) (upper surface) and Gamma(✓,↵) (lower surface), 0 < < 1, 4 < ✓ < 10. ]]> 0 is a real parameter and " = ("t , t 2 Z) a sequence of non-negative i.i.d. random variables. In order to assure the strict and weak stationarity of the processes X = (Xt, t 2 Z) and X2 = (Xt2, t 2 Z), we suppose that E(ln |"t|) and m8 exist and that 4 m4 < 1 , with mi = E("it), i 2 N (Gon¸calves, Martins and Mendes- Lopes, 2015). The nth moment of Xt, nX 4, can be expressed as n ✓n◆ where E(Xn"n) = 1 Xn ✓n◆ n i m E(Xn i"n i), tt 1 nmni=1i n+i t t E(Xn) = n i m E(Xn i"n i), tiitt i=0 n  4. It is easy to verify that E(Xt"t) = m2/(1 m1). The values E(Xtn"nt ), n = 1, 2, 3, are obtained recursively by using the previous equation; and finally, we getE(Xtn),forn4.Wenotethat 4m4 <1implies| nmn|<1,n=1,2,3, by Schwarz’s inequality. In this context, the Taylor property is only studied for h = 1, that is, by analyzing if ⇢X (1) > ⇢X2 (1), where ⇢X (1) and ⇢X2 (1) denote, respectively, the autocorrelations of lag 1 of the processes X and X2. It is enough to evaluate E(XtXt 1) and E(Xt2Xt2 1) in order to obtain these autocorrelations. Using (3.1) and the stationarity of the involved processes, we have E(XtXt 1) = E(Xt2"t) + E(Xt 1"t) = E( 2X2 "2 " +2 X " "2 +"3)+E(X " ). t 1 t 1 t t 1 t 1 t t t 1 t Taking into account the independence of the random variables "t, t 2 Z, and the strict stationarity of the related processes, we have E(X2 "2 " ) = 2 2 2 t 1 t 1 t m1E(Xt "t ) and E(Xt 1"t 1"t ) = m2E(Xt"t). Then E(XtXt 1) = 3m1E(Xt2"2t ) + 2 2m2E(Xt"t) + m1E(Xt) + m3. ]]> 8. In all cases, ↵ is a non-negative parameter and the condition E(|ln"t|) < +1 is satisfied. As in the case of the AR(1) model, the choice of these distributions takes into account the fact that the Taylor property seems to be related with the kurtosis value of the process. The uniform distribution is platykurtic with a constant kurtosis value equal to 1.8, while the exponential distribution is leptokurtic with a constant kurtosis value equal to 9. As we referred before, the kurtosis of the Pareto distribution is greater than 9. We also point out that, in all cases, the condition 4 m4 < 1 and the values of ⇢X(1) and ⇢X2(1) can be written in terms of r = ↵ . In each case, the value of the kurtosis of the process X given by (3.1) also depends on r = ↵ . We present the corresponding graphical representation. We point out that, in all these models, the leptokurtosis of the generator process implies the same property for the process X. In what concerns the Taylor property and kurtosis of X, comparisons are made separately between the first two distributions, uniform and exponential, and also between the Pareto E(Xt2Xt2 1"2t "2t 1) E(Xt2Xt 1"3t "t 1) E(XtXt2 1"t"2t 1) = 2m2E(Xt4"4t ) + 2 m3E(Xt3"3t ) + μ4E(Xt2"2t ), = 2m3E(Xt3"3t ) + 2 m4E(Xt2"2t ) + m5E(Xt"t), = m1E(Xt3"3t ) + m2E(Xt2"2t ), = m2E(Xt2"2t ) + m3E(Xt"t), E(XtXt 1"2t "t 1) E(Xt2"4t ) = 2m4E(Xt2"2t ) + 2 m5E(Xt"t) + m6, E(Xt"3t ) = m3E(Xt"t) + m4. Finally, the values of E(XtXt 1) and E(Xt2Xt2 1) appear in terms of the mo- ]]> 0.8 (approx.). We also observe that the Taylor property occurs for large values of the kurtosis, namely for KX (r) > 7.403 (' KX (1.1868987)). Generator process with exponential distribution with mean ↵ 41 Thecondition m4 <1isnowequivalentto08. ✓ In Figure 6, we present graphical representations oqf ⇢X (1) ⇢X 2 (1) and q4 ✓ 4 KX (r) for 0 < r < 0.85 and 9  ✓  50. We note that ✓ is an increasing functionof✓whoseminimumvaluefor9✓50is 4 5 '0.863. 4 ]]> 3. These graphical representations also suggest that the presence of the Taylor property is stronger for higher values of the kurtosis of the process X. In fact, as function of ✓, the di↵erence ⇢X (1) ⇢X2 (1) seems to increase when KX(r) increases, for all values of r that satisfy the condition 4m4 < 1. This situation strongly contributes to conjecture that the Taylor property and leptokurtosis are highly related in time series. 4. Conclusion The studies here presented confirm that linear and bilinear models may mimic the Taylor e↵ect. We note that in the linear case we were able to study the subject in a more complete way than in the bilinear class. In fact, a nec- essary and su cient condition assuring the presence of the Taylor property in the linear class for all lag h in the functions ⇢|X| and ⇢X2 was derived and we were able to conclude its presence in all models with symmetric or right-skewed distributed generator processes. New examples were studied and we still point out the strong relation between the Taylor property and the kurtosis of the process. In fact, in the platykurtic cases studied, the property is not present or is present in a very soft way, becoming more visible as the kurtosis goes away from the reference value of 3. In these cases, the presence of the property is not significant, in the sense ]]> 0, so-called because it was first n tt n ⇢ˆn (1) > ⇢ˆn (2) . (1.1) Ding, Granger and Engle [2], Granger and Ding [10], Granger, Spear and Ding [11] and Taylor [20] extend this initial analysis, finding that almost always, for n = 1, 2, ..., 625, ⇢ˆn (1) > ⇢ˆn (k) , for some values of k between 0 and 3. (1.2) So Granger and Ding [10] define the Taylor e↵ect to be present in a series if ⇢ˆn(1)>⇢ˆn(k), forallnandk6=1. (1.3) Granger [9] adds this to the list of stylized facts that characterize return se- ries dynamics. More studies with further discussion and evidence are cited in Haas [12]. Additionally, we refer that this e↵ect, more specifically the relation (1.1), has also been detect by Gon¸calves, Mendes-Lopes, Dorotoviˇc, Fernandes and Garcia [8] in physical phenomena time series. Taylor e↵ect strong empirical evidence raises the question whether or not empirically relevant volatility models, like the ones in the generalized autore- gressive conditional heteroskedastic (GARCH) class, can capture this e↵ect. In literature, two ways have been considered to address this problem: via simu- lation or using a theoretical approach. Simulations have the advantage of not requiring ⇢n (k) theoretical expressions for each model, however only a small number of model parameterizations can be considered. Since, in the GARCH ]]> ⇢n (2), for all n, (1.4) which they named Taylor property. He and Ter¨asvirta [13] consider the conditionally Gaussian first-order ab- solute value GARCH (AVGARCH) model and examine the Taylor property restricted to the autocorrelations of lag n = 1. They find the property to be present for some parameterizations associated with very high values of the model kurtosis. Gon¸calves, Leite and Mendes-Lopes [5] also evaluate this re- stricted version of the property, establishing its presence for some parameter- izations of the first-order threshold ARCH (TARCH) model but without pre- defining the generating process distribution. They also point to the influence the kurtosis of the generating process margins has on the parameterizations set which verify the property, since larger values seem to expand this set. Later, Haas [12] turns back to the first-order AVGARCH model, also not pre-defining the generating process distribution, and fully explores the Taylor property (for any lag n), extending the previous findings. The Taylor property has also been investigated with similar conclusions in stochastic volatility models, by Mora-Gal´an, P´erez and Ruiz [17], Veiga [21], P´erez, Ruiz and Veiga [18] and Malmsten and Ter¨asvirta [15], and also in bilinear models, by Gon¸calves, Martins and Mendes-Lopes [7]. In this paper, we consider the first-order power threshold GARCH ( -TGARCH) model, a general model that incorporates many popular GARCH models including the ones already mentioned, and discuss its ability to capture the Taylor e↵ect. Section 2 is dedicated to the presentation of the -TGARCH model. In Section 3, setting = 1, the presence of the Taylor property (for any lag) is established for parameterizations of the 1-TGARCH model that can reproduce the leverage e↵ects. Additionally, taking Student’s t-distribution with several degrees of freedom for the variables of the generating process, the influence of its kurtosis in this property is illustrated. Finally, in Section 4, an exploratory simulation study compares -TGARCH models regarding their ability to reproduce the Taylor e↵ect, considering several values of and Gauss- ian and Laplace distributions for the generating process margins. 2. The first order power threshold GARCH model Let X = (Xt,t 2 Z) and Z = (Zt,t 2 Z) be real stochastic processes. In the following, Xt+ = max {Xt, 0}, Xt = max { Xt, 0} and Xt 1 is the -field ]]>0,↵0 >0,↵1 0, 1 0, 1 0andZt isindependentofthe Xt 1. Z is called the generating process of X. If 1 = 0, we say that X follows a -TARCH(1) model. Regarding this specification of t, first proposed by Ding, Granger and En- gle [2], it is relevant to notice that it allows the model to capture the leverage e↵ect, being enough that ↵1 6= 1, and is more flexible since it does not estab- lish a priori the power value. Thus, it incorporates, for example, the spec- ificationsofGARCH( =2and↵1 = 1),AVGARCH( =1and↵1= 1), TGARCH ( = 1) and power-GARCH (↵1 = 1) models. In Gon¸calves, Leite and Mendes-Lopes [6] a detailed probabilistic analysis is presented, which in- cludes strict stationarity and stationarity up to -order of a more general version of this model, namely considering the -TGARCH(p,q) model with 6= 0 and assuming even less restrictive conditions for the generating process. As previously mentioned, the analysis of the Taylor property requires both ⇢n (1) = corr (|Xt| , |Xt n|) and ⇢n (2) = corr ⇣|Xt|2 , |Xt n|2⌘ analytical ex- pressions. Let He and Ter¨asvirta [13] proved that, for 2 {1,2} and k a positive integer, # ,k < 1 is a necessary and su cient condition for the existence of E ⇣|Xt| k⌘. Ling and McAleer [14] note that He and Ter¨asvirta’s proofs actually hold for any > 0 and that # ,k < 1 also guarantees the strict stationarity of X. However, only for = 1 both autocorrelations of interest were obtained in [13]. # ,k =E⇢h↵1 Zt+ + 1 Zt + 1ik . (2.2) Thus, in the following, let us assume that X is a 1-TGARCH(1,1) (also writ- ten TGARCH(1,1)) process such that the generating process margins have zero mean, unit variance and a symmetric distribution with finite fourth moment. To simplify our notation, let #k = #1,k and k = E ⇣|Zt|k⌘. If #2 < 1, X is strictly and weakly stationary and ⇢n (1) exists for all n, with ⇢1(1)=#1 + 1 1 #21 21 1 (2.3) (1 #21) 21 (1 #2) ]]> 1, ⇢ (1)=#n 1⇢ (1). (2.4) n11 The su cient condition for the existence of the squared process autocorre- lations is #4 < 1, which implies #2 < 1. Under this condition, ↵ 12 + 12 3 12 ⌥ ⇢1(2)= 2 +(↵1+ 1) 1 + +(1 #4) (2.5) (2.6) and, for n > 1, where ⌥=[(↵1 + 1) 3 +2 1](1+2#1 +2#2 +#1#2)(1 #1)(1 #2) n 1 ⇢ (2)=#n 1⇢ (2)+(1 # ) ⇤ X#n i 1#i, n214 21 i=1 44 + ⇣ ↵ 21 + 12 + ( ↵ 1 + 1 ) 1 3 + 12 1 ⌘ ( 1 + # 1 ) 2 ( 1 # 3 ) + 1 # 21 ( 1 # 2 ) ( 1 # 3 ) , 2 4 4 ⇤ = 2#1 (1+#1) (1 #3) + [(↵1+ 1) 3 + 2 1] (1+2#1+2#2+#1#2) (1 #1) , = 4 (1 #1) (1 #2) (1 + #1)2 (1 #3) (1 #4) , and =1+3#1 +5#2 +3#3 +3#1#2 +5#1#3 +3#2#3 +#1#2#3. We note that the parameter ↵0 has no interference in the existence conditions nor in the autocorrelations expressions. The expressions of the autocorrelations displayed here have been suitably adapted from [13]. These adaptations were crucial in obtaining the results pre- sented in the following section. 3. The Taylor property in the 1-TGARCH(1,1) model In this section, we begin by establishing the presence of the Taylor property in the 1-TGARCH(1,1) model. Then, we explore the extent of the parameteriza- tions region that verifies it. Finally, considering the first-order autocorrelations and particular distributions for the generating process margins, we illustrate the influence of its kurtosis in the size of the referred region. To develop this study we assume an additional condition on the moments of the generating process, more specifically, that 1/4 < < 1. This fundamen- tal relation, first introduced by Gon¸calves, Leite and Mendes-Lopes [5], was further analysed and then characterized by Haas [12], as “will be satisfied for pratically any distribution that one can anticipate in the context of GARCH models”. 41 ]]> 1/4, we can state that 1/4 > 1414 1/2 and conclude that there exists a neighbourhood of ( 1/4, 1/4, 0) that, 444 intersected with C, only has points corresponding to parameterizations of the 1-TGARCH(1,1) that verify ⇢1(1) > ⇢1(2). 144 In a similar manner, for each n in N\ {1}, when (↵1, 1, 1) converges to ( 1/4, 1/4, 0) through values in C, 44 ⇣ 1⌘n ⇢ (1) = #n 1⇢ (1) ! 4 , n1114 ⇤X ⇣⌘n n 1 1 ⇢ (2)=#n 1⇢ (2)+(1 # ) #n i 1#i ! 2 , n214 214 i=1 and so we can conclude that there exists a neighbourhood of ( 1/4, 1/4,0) 44 such that, intersected with C, its elements satisfy ⇢n (1) > ⇢n (2). ⇤ ]]> 0 such that 1+k✓ 2 ◆ 1/4 1 1 + k2 k4 + 1 > 1/4 . (3.1) 14 As 1/4 < is equivalent to ( 1/4) 1 < 1, then the solutions k > 0 of 4114 (3.1) are also solutions of the following inequality 1+k ✓ 2 ◆ 1/4 1 + k2 k4 + 1 1, (k 1)2 k6 +2k5 +3k4 +8k3 +3k2 +2k 1 0. (3.2) (3.3) which is equivalent to Numerical analysis methods allow us to determine that if k 2 [0.2814, 3.5546] then it satisfies (3.3), thus also satisfies (3.1). In view of the previous calculations, we can conclude that the Taylor prop- erty holds, at least, for parameterizations in a neighbourhood of the boundary of existence of the autocorrelations such that 0.2814↵1 < 1 < 3.5546↵1 (so ↵1 and 1 can be clearly di↵erent) and with 1 close or equal to zero, covering ]]> ⇢1 (2) for the 1-TARCH(1) model. In Figure 1 we can observe that the region where ⇢1 (1) > ⇢1 (2) is verified seems to enlarge as the kurtosis, kZt , of the generating process variables in- creases, since we have, for the decreasing values of the considered degrees of free- dom, kZt equal to 3.05, 3.23, 3.55, 3.6, 5 and 9, respectively. It is also relevant to notice that when kZt is equal 5 and 9, almost all, if not all, parameterizations for which the existence of the autocorrelations is ensured, verify ⇢1 (1) > ⇢1 (2). These representations were obtained for other centered and reduced distribu- tions, namely, triangular (kZt =2.4), Gaussian (kZt =3) and Laplace (kZt =6), displaying the same behaviour. 4. The Taylor effect in the -TGARCH(1,1) model In this section, we consider the first-order -TGARCH model with not necessarily equal to 1. Since both theoretical expressions of ⇢1 (1) and ⇢1 (2) are only available for = 1, the Taylor e↵ect is analysed here via simulation. Taking into account the final remarks of the previous section, we considered two zero mean and unit variance distributions for the generating process mar- gins, namely the Gaussian and Laplace. For these distributions and in order to compare the ability that models with di↵erent values of have to gener- ate series with the Taylor e↵ect, we considered four values of , specifically 0.25, 0.5, 1 and 2, and three parameterizations for each one of them. In order to guarantee the existence of the autocorrelations, all parameterizations satisfy # ,4/ < 1, with 4/ an integer. These eight sets of three parameterizations have (n 2)⇡ n n 2 2 ]]> TARCH(1) model with generating process marginal distribu- tion based on Student’s t-distribution with df degrees of free- dom. approximately the same values of the model theoretical kurtosis. Our choice of comparing the models with the same kurtosis is related to the findings of He and Ter¨asvirta [13] for the AVGARCH(1,1) model, who calculated the theoreti- cal kurtosis expression and observed that its high values favours the appearance of the Taylor e↵ect. For each parameterization, 100 trajectories with 4000 ob- servations were obtained, with the first 1000 observations discarded to remove the e↵ect of the initial values choice. For each simulated trajectory, the sam- ple autocorrelation of the absolute observations was compared with the one of squared observations. The simulation results are presented in Table 1 for the Gaussian distribution and in Table 2 for the Laplace distribution. ]]> ⇢ˆ1 (2) Score (out of 100) 11.73 91 0.545 0.545 6.94 93 2 0.7 0.1 0.53 0.53 0.3 0.686 0.68 0.1 0.44 0.44 11.60 58 6.57 66 38.51 28 11.21 11 6.62 9 Table 1. Simulation results considering the -TGARCH(1,1) model with ↵0 = 1, 1 = 0.1, Zt ⇠ Gaussian and , ↵1, 1 specified above and K is the model theoretical kurtosis. ↵1 1 K ⇢ˆ1 (1) > ⇢ˆ1 (2) Score (out of 100) 0.25 0.46 0.35 0.2 0.5 0.2 0.4 0.2 1 0.2 0.47 0.2 2 0.1 0.35 0.15 Table 2. Simulation results considering the -TGARCH(1,1) model with ↵0 = 1, 1 = 0.1, Zt ⇠ Laplace and , ↵1, 1 specified above and K is the model theoretical kurtosis. 0.6 0.2 0.2 0.62 0.05 0.2 0.645 0.1 0.2 38.73 99 11.44 99 6.91 98 38.82 93 11.94 83 6.73 95 37.59 82 11.2 78 6.68 79 0.5 0.19 0.15 6.82 35 33.6 45 11.25 31 ]]>0, if ⇠6=0, (1.1) exp[ exp( x)], x 2 R, if ⇠ = 0. We then say that F is in the domain of attraction for maxima of EV⇠, denoting this by F 2 DM (EV⇠ ). The EV⇠ d.f., in (1.1), incorporates the three Fisher-Tippett types: the Gum- bel family, ⇤(x) ⌘ EV0(x) := exp( exp( x)), x 2 R, (⇠ = 0), the limit for exponentially-tailed distributions; the Fr´echet family, ↵(x) ⌘ EV1/↵(↵(x 1)) := exp( x ↵), ↵ > 0, x > 0, (⇠ = 1/↵ > 0), the limit for negative polynomial heavy-tailed distributions and the Weibull family, ↵(x) ⌘ EV 1/↵ (↵(x + 1)) := exp( ( x)↵), ↵ > 0, x < 0, (⇠ = 1/↵ < 0), the limit for short-tailed distributions. If ⇠ = 0, the right endpoint, x⇤ := sup{x : F(x) < 1}, can then be either finite or infinite. If ⇠ > 0, F has an infinite right endpoint. If ⇠ < 0, F has a finite right endpoint. The shape parameter, ⇠, is directly related to the weight of the right tail, F := 1 F , of the underlying model F . As ⇠ increases the right tail becomes heavier. Whenever independence is no longer valid, some important dependent se- quences have been studied. The limit distributions of their order statistics, under some dependence structures, have been obtained. Stationary sequences are examples of such sequences and are realistic for many real problems. As dependence in stationary sequences can assume several forms, some con- ditions have to be imposed. The first condition, known as the D(un) dependence condition, Leadbetter et al. (1983), ensures that any two extreme events can become approximately independent as n increases when separated by a rela- tively wide interval of length ln = o(n). Hence, D(un) limits the long-range dependence between such events. Under adequate local dependence conditions, the limiting d.f. of the max- imum of the stationary sequence may be directly related to the maximum of the i.i.d. associated sequence, through a new parameter, the so-called extremal index, usually denoted by ✓. This parameter also a↵ects other parameters of extreme events, so its reliable estimation is of great importance. However, estimators of ✓ show the usual behaviour of semi-parametric tail estimators: nice asymptotic properties, but a high variance for small k, the ]]> 0, the underlying d.f. F verifies F(un) = 1 ⌧/n + o(1/n), n ! 1. (2.1) A stationary sequence {Xn}n 1 from an underlying model F is said to have an extremal index ✓ (0 < ✓  1) if, for all ⌧ > 0, we can find a sequence of levels un = un(⌧) such that with {Yn}n 1 the associated i.i.d. sequence (i.e., an i.i.d. sequence from the same F ), it is verified P(Yn:n  un) = Fn(un) ! e ⌧ and P(Xn:n  un) ! e ✓⌧, n!1 n!1 (see Leadbetter et al. (1983)). ]]> 0} and {bn }, if and only if, P r {(Mn bn )/an  x} ! G2(x) where G2(x) = G✓1(x), for a constant ✓ such that 0 < ✓  1. n!1 So, given that G1(·) ⌘ EV⇠(·), the limit law G2(·) ⌘ EV⇠✓(·) is an extreme value d.f. with location, scale and shape parameters (μ✓, ✓,⇠✓) given by 1 ✓⇠ ⇠ μ✓=μ ⇠ , ✓= ✓ and ⇠✓=⇠, where (μ, , ⇠) are the location, scale and shape parameters, respectively, of the limit law of the i.i.d sequence. The extremal index ✓, 0 < ✓  1, is directly related to the clustering of ex- ceedances: ✓ = 1 for i.i.d. sequences and ✓ ! 0 whenever dependence increases. Definitions can be given extending values of ✓ to the case ✓ = 0, but this is a situation out of interest, see Beirlant et al. (2004). For illustration of the behaviour of a stationary process for some values of ✓ let us consider the following examples: Example 2.2. Let {Xn}n 1 be the two-dependent sequence defined as Xn = max(Zn+1,Zn), n 1, where {Zi}i 1 are standard exponential i.i.d. r.v.’s. The underlying model for Xn is given by F (x) = (1 exp( x))2, x 0. Let {Yn}n 1 be a sequence of i.i.d. r.v.’s from the same distribution, i.e., F(y) = (1 exp( y))2, y 0. Figure 1 shows a size equal to 2 for the clusters of exceedances of high levels by the {Xn} sequence. Actually for that sequence we have ✓ = 1/2. It can also be seen a shrinkage of the largest observations for the 2-dependent sequence, despite of the fact that we have the same model underlying both sequences. Example 2.3 (Beirlant et al., 2004, Example 10.3). “Max-Autoregressive Pro- cess (ARMAX process)”. ]]>un]= limPr[X1 un|X2 >un]. (2.2) n!1 n!1 Given a sample, (X1, . . . , Xn), the empirical counterpart of the above inter- pretation led to the naive estimators: the classical up-crossing (down-crossing) estimator, UC estimator , ⇥bUC (DC-estimator, ⇥bDC) (Nandagopalan, 1990; Gomes, 1990, 1992, 1993) n 1 P I(Xi un un) for a suitable threshold un, where I(A) denotes, as usual, the indicator function of A. Consistency of this estimator is obtained provided that the high level un is a normalized level, i.e. if with ⌧ ⌘ ⌧n > 0, the underlying d.f. F verifies F(un)=1 ⌧/n+o(1/n), n!1 and ⌧/n!0. Di↵erent forms of identifying clusters gave rise to other estimators. Let us men- tion two very popular estimators: the blocks estimator and the runs estimator, Hsing (1991, 1993). The blocks estimator is derived by dividing the data set into approximately kn blocks of length rn, where n ⇡ kn ⇥ rn, i.e., considering kn = [n/rn]. Each block is treated as one cluster and the number of blocks in which there is at least one exceedance of the threshold un is counted. The blocks estimator, ⇥bBn (un), is then defined as Pkn I max X ,···,X >u ⇥bB(u):= i=1 P(i 1)rn+1 irn n. n n ni=1 I (Xi > un) If we assume that a cluster consists of a run of observations between two exceedances, then the runs estimator is defined as: Pn i=1 n 1 P I(Xi >un,Xi+1 un) ⌘ i=1 := ⇥bDC(un) I(Xi > un) (2.3) ⇥n(un):= Pni=1I(Xi >un) . Pn i=1 bR Pni=1 I Xi > un,max Xi+1,··· ,Xi+rn 1  un Under mild conditions, limn!1 ⇥bBn (un) = limn!1 ⇥bRn (un) = ✓. Other proper- ties of these estimators have been well studied by Smith and Weissman (1994) and Weissman and Novak (1998). ]]>0. ky xkh Of course, if q is continuous at x then limh!0 wq (x, h) = 0. 2. Assumptions on small-ball probabilities We introduce a first assumption on the behaviour of small-ball probabilities. (SB1) Let F(x,h) = P(kx Xk  h), and assume that, for each x 2 S, limh!0 F (x, h) = 0 and F is partially di↵erentiable with respect to h. ]]> 0. Thus, kernels with weights decreasing smoothly to 0, such as K(z) = 1 z✓, ✓ > 0, are not allowed under the assumptions in [6]. The need for such extra asumptions on K is due to the peculiarities of the control of the asymptotics, made through distribution functions and not needing the convergence rate towards ⌧0. However, kernel functions satisfying the assumptions of [6] leave us with an estimator the resambles the regresso- gram, as there is a sudden cut-o↵ on the weights once the observation becames to far from the reference point x. Examples 2.1. We shall next describe in more detail the functions and in a few significative examples. The models (1)–(3) are also discussed in Ferraty, Mas, and Vieu [6] describing the characterizations adapted to their assumptions. (1) Let F(x,h) = f(x)h , > 0. If 2 N this model includes the case of X being a -dimensional random vector. For general > 0, the model corresponds to fractal type processes of order (see Ferraty and Vieu [3, 5]). It is easily verified that F0(x, z) = z 1, (x, z, h) = 0, thus, also (K, x, h) = 0. (2) With respect to the previous example, we may relax the concentration rate of the probability mass around x assuming F (x,h) = f (x)h |log h|k , , k > 0. Then, F0(x,z) = z 1, (x,z,h) = kz 1 , and K2(z)F0(z) dz. |log h| kZ1 (K, x, h) = |log h| K(z)z 1 dz ! 0, 0 if this last integral is finite. Thus (K,x,h) does converge to 0, but with a rather slow rate. (3) Assume F(x,h) = f(x)h e h k, , k > 0. This model for the distri- bution of X includes di↵usion processes, fractional Brownian motions, fractional Brownian sheets or fractional Ornstein-Uhlenbeck processes. ]]>t}⇠'(1),where' t is a slowly varying function. In this case, it has been shoRwn recently by Karol and Nazarov [12] that P(kXk  h) ⇠ exp⇣ 1 u '(z) dz⌘, 21z where '(u) ⇠ h. Assume that n ⇠ n ↵, ↵ > 1 (the Brownian motion 2u corresponds to ↵ = 2). Then '(z) = z1/↵ and it is easily verified that 1 ↵ 1 ↵ P(kXk  h) ⇠ e↵/2 exp⇣ 22↵ 1 ↵h ↵ ⌘, which corresponds to model (3) with = 0 and k = ↵ . ↵ 1 3. Convergence of the estimator Using Lemma 2.1 in [10] it follows that, for " > 0 small enough, n b o rb n ( x ) E gb n ( x ) > " n E f n ( x ) o n ( E fb ( x ) ) 2 o (3.1) ⇢ |gb(x) Egb(x)|>"Efb(x) [ fb(x) Efb(x) >" n . n n 4n n n 4Egbn(x) ]]>0 such that, for every ` 2, E(Y`|X =x)M1``!s(x) and nF (x, h) ! +1, (3.3) log n then rbn(x) E rbn(x) converges almost completely to zero with a convergence rate of order ⇣ log n ⌘1/2. nF (x,h) Theorem 3.6. Assume r is continuous, (SB1), (SB2) and (K) hold. Then p 0 f1(x)+ lim (K2, x, h) 1 d @ h!0 2 A nF(x,h)(rbn(x) Erbn(x)) !N 0, f0(x)+lim (K,x,h) (s(x) r (x)) . h!0 4. The mean square error We want to characterize the asymptotic behaviour of E (rbn(x) r(x))2. For this, as done in Bosq and Cheze [2] and later explored by Oliveira [15] and Bensa¨ıd and Fabre [1], we consider the decomposition E(rb (x) r(x))2 = r2(x)E(fb (x) f (x))2 n f02(x)n0 + 1 E ( gb n b( x ) r ( x ) f 0 ( x ) ) 2 f 02 ( x ) 2r(x) E ⇣(fn(x) f0(x))(gbn(x) r(x)f0(x))⌘ f 02 ( x ) (4.1) ⇣2 2 bb 2⌘ f02(x)n n0 1 + E (rb (x) r (x))(f (x) f (x)) 2 E ⇣(rbn(x) r(x))(fn(x) f0(x))(gbn(x) r(x)f(x))⌘ . f 02 ( x ) ]]>"n } ⇣22b2⌘ +E (rb (x) re (x))(f (x) f (x)) I =:A +B +C . n n n 0 {rbn(x)>↵n} nnn Lemma 4.4. Assume that (SB1), (SB2) and (K) hold. If ↵n % +1, "n & 0 are such that ↵n"n ! 0 then An =o✓ 1 + 2(K,x,h)◆. nF (x, h) Proof. The result is obvious by noting that A  (↵ + re (x))" E (fb (x) nnnnn f0(x))2 and taking into account Lemma 4.1. ⇤ To bound Bn, we start by remarking that B (↵ +re (x))2⇣E(fb(x) f (x))4⌘1/2(P(|rb (x) re (x)|>" ))1/2, nnnn0nnn and using (3.1) to obtain P(|rbn(x) ren(x)| > "n) ⇣ b ⌘ ⇣ b b P |gbn(x) Egbn(x)|>"nEfn(x) +P fn(x) Efn(x) >"n ( E fb ( x ) ) 2 ⌘ n . ◆, 4 4 E gb n ( x ) Using now Markov’s inequality it follows that P⇣|gb(x) Egb(x)|>"nEfb(x)⌘ 16Vargbn(x) = 1O✓ 1 n n 4 n " 2 ( E fb ( x ) ) 2 " 2n n F ( x , h ) nn and analogously for the other probability term. Lemma 4.5. Assume that (SB1), (SB2), (K) and (3.2) hold. Then Bn =o✓ 1 + (K,x,h) + 2(K,x,h)◆. nF (x, h) n1/2 F 1/2 (x, h) ]]> " ))1/2 nn0nnn ↵n2✓1 (K,x,h)2 ◆✓1◆ = " O nF(x,h)+n1/2F1/2(x,h)+ (K,x,h) O n1/2F1/2(x,h) . n Nowchoose↵, >0suchthat2↵+ <1 andput↵n=(nF(x,h))↵and"n= 2 = (nF(x,h))2↵+ 1/2 ! 0, so the term Lemma 4.7. Assume that wr(x,·), ws(x,·), (K,x.·) and (K2,x,·) are bounded, (SB1), (SB2), (K) and (3.2) hold. If, for some ↵ > 0, (nF(x,h)) . Then ↵2n "n n1/2 F 1/2 (x,h) ⌘ + 2(K, x, h) . The term corresponding ⌘ ⇣ to P fb (x) E fb (x) > (E fb (x))2 above is an o 1 + (K,x,h) nF (x,h) n1/2 F 1/2 (x,h) ⇣ is treated analogously. ⇤ Remark 4.6. If we choose > ↵ then ↵n"n ! 0. "n n n n 4 Egbn(x) then 4 E(rb(x)I n ↵ ) !0, { rb n ( x ) > ( n F ( x , h ) ) } + (K,x,h) + 2(K,x,h)◆. n1/2 F 1/2 (x, h) (4.2) Cn =o✓ Proof. Put ↵n = (nF (x, h))↵ ! +1 and write 1 nF (x, h) ⇣bb⌘ 222 Cn = E (rbn(x) ren(x)(x))(fn(x) f0(x)) I{rbn(x)>↵n} ⇣  E (fn (x) f0 (x)) E (rbn (x)I{rbn (x)>↵n } ) b 4⌘1/2 4 1/2 ⇣ 4⌘1/2 2 1/2 E (fn(x) f0(x)) ren(x) (P(rbn(x) > ↵n)) . ⇤ + The result is now obvious taking into account (4.2). To complete the characterization of the mean square error we still have to control the final term in (4.1). The asymptotic behaviour of this last term depends more directly on the geometry and smoothness the distribution of X. ]]>↵>0and2↵+ < 1.Notethat,as↵< 1,thischoiceisalwayspossible. 26 Decompositing as before, the first term in (4.3) is bounded above by h⇣2b2⌘ (↵n + ren(x)) E (rbn(x) ren(x))b(fn(x) f0(x)) I{rbn(x)↵n,|rbn(x) ren(x)|"n} +E ⇣(rb (x) re (x))2(fb (x) f (x))2I ⌘i n n n 0 {rbn (x)↵n ,|rbn (x) ren (x)|>"n } +E ⇣(rbn (x) ren (x))2 (fn (x) f0 (x))2 I{rbn (x)>↵n } ⌘ . ]]>↵n}⌘ ⇣222⌘ 2E (rb (x)+re (x))(f (x) f (x)) I n n n 0 { rb n ( x ) > ↵ n } = o ✓ 1 + 2(K, x, h) + (K, x, h) ◆ , nF (x, h) n1/2 F 1/2 (x, h) taking into account (4.2). Collecting these upper bounds, the result follows. ⇤ To describe the mean square error for the regression estimator it remains to put together the characterizations just obtained for each term of (4.1). Theorem 4.9. Assume that r is continuous, ws(x, ·), (K, x.·) and (K2, x, ·) are bounded, (SB1), (SB2), (K), (3.2) and (4.2), for some ↵ 2 (0, 1 ), hold. Then, E(rbn(x) r(x))2 = f1(x) + (K2, x, h) ✓ r2(x) + s(x) + ws(x) 2r(x) ◆ 6 nF (x, h) f02 (x) f02 (x) f02 (x) +wr2(x,h)(f0(x)+ (K,x,h))2 2r(x)wr(x,h) (K,x,h)(f0(x)+ (K,x,h)) +o f02 (x) f02 (x) 1 + 1/2(K,x,h) + (K,x,h)+ 1/2(K,x,h)+wr(x,h) nF (x, h) n3/4F 3/4(x, h) n1/2F 1/2(x, h) + 3/2(K,x,h)+ 1/2(K,x,h)wr(x,h)+ (K,x,h) n1/4 F 1/4 (x, h) + 2(K,x,h)+ 3/2(K,x,h)+ (K,x,h)wr(x,h)!. Let us get back to the case where limh!0 (K, x, h) = 0. We may trace the calculations above to find that, under this additional assumption and the ]]>R Here, (Zt) constitutes a sequence of integer-valued random variables (r.v’s), and R represents the threshold level. The “↵i ” is the binoPmial thinning operator of Steutel and van Harn (1979). It is defined as ↵ X := Xi=1 Yi, for X with range N0 = {0, 1, . . .}, where the (i.i.d.) Bernoulli variables with probability ↵ 2 (0; 1). The authors discussed probabilistic and statistical properties related with this class of models. Note that the SETINAR models fall within the state-dependent thinning class. In this article, we introduce the periodic self-exciting threshold integer- valued autoregressive model of order one with two regimes (hereafter referred to as PSETINAR(2; 1, 1)T ) which generalizes the SETINAR model by considering periodically varying threshold levels. For this class of models, we investigate some basic probabilistic and statistical properties. Furthermore, parameter es- timation and forecasting are also addressed. Finally some concluding remarks are given. The PSETINAR(2; 1, 1)T model is defined through the recursive equation Yi’s are independent and identically distributed ( ↵(1) X + Z(1), X  R Xt = j t 1 t t d t , t 2 N0, (1.1) ↵(2) X + Z(2), X > R j t 1 t t d t withRt =rj,fort=j+sT,j=1,...,Tands2N0.Notethatforthe jth-period we have Xj+sT = (↵(1) Xj+sT 1 + Z(1) )I(1) + (↵(2) Xj+sT 1 + Z(2) )I(2), j j+sT j j j+sT j with I(k), for k = {1, 2⇢}, defined as j I(1) := 1, Xj+sT d rj , I(2) =1 I(1). j 0, Xj+sT d >rj j j (1.2) (1.3) The threshold parameter Rt (which is assumed to be known) represents the level of the process and the regime switch is triggered by the lag-d value of the series. ]]>rj j P2 P2 P2 h (k1) (k2) (kj) (k1,k2,...,kj) ! (k1,k2,...,kj)i E[Xt] = ··· ↵1 ↵2 ···↵j ⇥u1,2,...,j ⇥p1,2,...,j fort=j+sT,j=1,...,T ands2N0. 3. Parameters estimation In this section, we consider the parameter estimation of the PSETINAR(2; 1, 1)T process. In particular, the conditional least squares (CLS) and conditional max- imum likelihood methods are adopted. For this purpose, let (X1, . . . , Xn) be a sequence of r.v’s satisfying (1.4) and denote by ✓ := (↵(1),↵(2), ,...,↵(1),↵(2), ), 111TTT the vector of unkown parameters. Recall that Rt is assumed to be known. 3.1. Conditional least squares estimators. The CLS-estimators ,↵ˆ(2) , ˆT,CLS), T,CLS k1 =1k2 =1 kj =1 Pj P2 P2 ( k l + 1 ) ( k j ) k l + 1 , . . . , k j + l ··· ↵l+1 ···↵j ⇥pl+1,...,j , l=1 kl+1 =1 kj =1 ✓ˆCLS := (↵ˆ(1) ,↵ˆ(2) , ˆ1,CLS,...,↵ˆ(1) 1,CLS 1,CLS are obtained by minimizing the expression T,CLS ◆2 with N and T denoting the number of complete cycles and number of periods, N 1 T ✓ Q(✓):= XX Xj+sT gj(✓j,Xj+sT 1) s=0 j=1 respectively. Moreover, ✓j := ⇣↵(1),↵(2), j⌘ and the function gj takes the form jj gj(✓j,Xj+sT 1)=↵(1)Xj+sT 1I(1) +↵(2)Xj+sT 1I(2) + j. jjjj Solving the systems of 8the form @Q > > > >: ( 1 ) = 0 >< @ ↵ j > @↵j @Q = 0, (2) j = 1,...,T, (3.1) @Q =0 @ j ]]> > > > ↵ˆ ( 1 ) > j,MQC > j,MQC P = ◆2 Xj+sT 1Ij N Xj+sT Xj+sT 1Ij Xj+sT s = 0 s = 0 s = 0 N 1 (1) N 1 N 1 PPP (1) Xj+sT 1Ij > > N 1 ✓N 1 N P X2 I(1) P Xj+sT 1I(1) s=0 j+sT 1 j s=0 j N 1 (2) N 1 N 1 >

N Xj+sT Xj+sT 1Ij Xj+sT >↵ˆ(2) = s=0 s=0 s=0 (2) >: ˆj,MQC =N 1 PXj+sT ↵(1) PXj+sT 1I(1) ↵(2) PXj+sT 1I(2) jjjj s=0 s=0 s=0 for j = 1,...,T. The consistency and asymptotic distribution of the CLS- estimators are established in the result given below. Theorem 3.1. The CLS-estimators are strongly consistent and asymptotically ◆2 >✓◆ ✓P , N 1 > (2) (2) s=0 s=0 > N 1 N 1 N 1 > > N 1 (N Xj2+sT 1Ij Xj+sT 1Ij normal, i.e., pˆd 1 1 n(✓ ✓) !N(0,V WV ), (3.2) where V and W are square matrices of order 3T defined by blocks of 3 ⇥ 3 given by 26 : 1 0 . . . 0 37 26 ⌦ : 1 0 . . . 0 37 6 0 :2 ... 0 7 6 0 ⌦:2 ... 0 7 V=64 . . ... . 75 and W=64 . . ... . 75, (3.3) where 0 0 ... :T 0 0 ... ⌦:T (k,l):j := E @ gj(✓j,Xj+sT 1) @ gj(✓j,Xj+sT 1) , @✓k:j @✓l:j :=EU2 @ g(✓,X ) @ g(✓,X ) , k,l 2 {1,2,3}; j = 1,...,T and ✓j := (✓1:j,✓2:j,✓3:j) ⌘ ⇣↵(1),↵(2), j⌘, are ⌦ are the elements of the matrices :j and ⌦:j , j = 1, . . . , T , respectively, with (k,l):j j+sT @✓a:j j j j+sT 1 @✓l:j j j j+sT 1 jj the parameters associated to j-th period. Proof. Consistency and asymptotic normality can be proved using the results of Klimko and Nelson (1978, sec.3). It is easily checked that all regularity con- ditions by Klimko and Nelson (1978, p. 634) are satisfied, and thus, by their ]]>rj] jhijhi m(1) := E Xi |Xj 1+sT d  rj m(2) := E Xi |Xj 1+sT d > rj j,i j 1+sT j,i j 1+sT q(1) := P [Xj+sT d  rj] q(1) := P [Xj+sT d > rj]. jj Noticing that det| :j| =6 0, 8j = 1,...,T lead us to conclude that V is in- vertible and condition (C) is thus fulfilled. Thus, Theorem 3.2 of Klimko and Nelson (1978) is thereby satisfied implying that the CLS-estimators are asymp- totically normal. This concludes the proof. ⇤ 3.2. Conditional maximum likelihood estimators. For a fixed value of x0, the conditional likelihood function for the PSETINAR(2; 1, 1)T takes the form N 1 T QQ s=0j=1 ⇣ N 1 T QQ s=0 j=1 L(✓) := = Pj (Xj+sT = xj+sT |Xj 1+sT = xj 1+sT ) ⌘ ⇣⌘ x m (1) (1) (2) (2) pj xj 1+sT,xj+sT,↵j Ij +↵j Ij , j with = pj (xj 1+sT , xj+sT )(1) + pj (xj 1+sT , xj+sT )(2) andM⇤ :=min(xj 1+sT,xj+sT). The CML-estimators ✓ˆCML :=(↵ˆ(1) ,↵ˆ(2) , ˆ1,CML,...,↵ˆ(1) ,↵ˆ(2) , ˆT,CML), 1,CML 1,CML T,CML T,CML pj ⇣xj 1+sT,xj+sT,↵(1)I(1) +↵(2)I(2), j⌘= jj jj M⇤ 2 j XX xj 1+sT (k)m (k) (xj 1+sT m) j+sT (k) Cm ↵j 1 ↵j (xj+sT m)!Ij = e ⌘ pj xj 1+sT , xj+sT , ↵(1)I(1), j + pj xj 1+sT , xj+sT , ↵(2)I(2), j ⇣ m=0k=1 ⌘ jj jj ⇣ ⌘ ]]> > > > N 1 I(k) X (k) j (x x↵) ↵(k)(1 ↵(k)) j+sT j 1+sT j > < p ⇣j j s=0 ⌘ x ,x 1,↵(1)I(1) +↵(2)I(2), > > j j j 1+sT j+sT j j j j j p ⇣x ,x ,↵(1)I(1) +↵(2)I(2), ⌘ > j j 1+sT j+sT j j j j j > >: P pj s=0 jjjj N 1 ⇣ (1) (1) (2) (2) ⌘ xj 1+sT ,xj+sT 1,↵j Ij +↵j Ij , j pj ⇣xj 1+sT ,xj+sT ,↵(1)I(1)+↵(2)I(2), j ⌘ = N for k = 1,2 and j = 1,...,T. In order to solve those systems numerical proce- dures have to be employed. Note, however, that the CML-estimates for the ’s, are readily available from those for the ↵’s through the following expression 1 N 1 X⇣⌘ s=0 xj+sT ↵ˆ(k) xj 1+sT , j=1,...,T. N j,CML ˆj,CML = The following result establishes the consistency and the asymptotic distribution of the CML-estimators. Theorem 3.2. Let (Xt ) be the PSETINAR(2; 1, 1)T model in (1.1). The CML- estimators are asymptotically normal, i.e, ↵ˆ ( 2 ) ↵ ( 2 ) 2 11 M0...0 3 ˆ1 1 6 1 7 pn . !d N(0,I 1), where I=6 0 M2 ... 0 7 , . 4 . . ... . 5 ↵ˆ ( 1 ) ↵ ( 1 ) T T 00...MT ˆT T ↵ˆ ( 2 ) ↵ ( 2 ) TT ]]> i n 1 0i 1 1 @AX@A j=0 i=1 j=0 n 1 ⌘ t,n Xt n + t,i Zt i, i=0 1 d Xt = t j Xt n + t j Zt i + Zt ⇢ Qi 1 t j, i>0 ⇢ t,j T`,T, i=j+`T, j=1,...,T t,i:= j=0 = , 1, i=0 1, i=0 leading to obtain d hX 1 Xi+NT+h = i+NT+h,h Xi+NT + i+NT+h,m Zi+NT+h m. (4.1) m=0 Since h = j + `T, it follows that `T 1 ` 1 T 1 X j+`T,m Zj+`T m =d X X j+`T,m T`,T Zj+(` w)T m, m=0 w=0 m=0 the expression in (4.1) takes the form Xi+NT+h =d i+j+(N+`)T,j+`T Xi+NT + Due to the periodicity of the ’s, i+j+(N+`)T,j+`T = i+j,j+`T = i+j,j T` ,T , and considering the relation with Xi+NT+h =d i+j,j T`,T Xi+NT +Vi+j+`T i 1 ` 1 T 1 (4.2) Vi+j+`T := X i+j,m Zi+j m+NT +X X i+j+(N+`)T,m+j+Tw Zi+(N+` w)T m. m=0 w=0 m=0 j + X` T 1 m=0 i+j+(N+`)T,m Zi+j+(N+`)T m. ]]> m; i+j m i+j m p(1):=PX i m i+(N +` w)T m 1 p(2) =1 p(1) , i>m. i m i m r ; i m Turning now to the particular case h = 1, the one-step-ahead predictive func- tion is given by P(Xj+NT+1 = y|Xj+NT = x) ⇣ ⌘ ⇣ ⌘ (y m)! with (k) = E[Z(k) ], k 2 {1,2}. Finally, from (4.2), the most commonly used one-step-ahead predictor of Xj+NT+1, takes the form Xˆj+NT+1 = ⇣↵(1) Xj+NT + (1) ⌘P(Xi+NT ri+1) + ⇣↵(2) Xj+NT + (2) ⌘P(Xi+NT >ri+1). j+1 j+1 j+1 j+1 ]]> 0, fYt/Xt(·/x) ⌘ f(·/x) and FYt/Xt(·/x) ⌘ F(·/x) denote the density and the distribution function of Yt given Xt = x, respectively. We also consider q↵(x)=inf{z2R:F(z/x) ↵}, ↵2]0,1[, the conditional quantile function of Yt given Xt = x. Our estimator of FZ(·/x) is defined by +1 d FT(y/x)= 1] 1,y](z)fT(z/x)dz, y2R, x2R , 1 where fT (·/x) is the kernel estimator of the conditional density fT (·/x) pro- posed by Didi [13]: 1 Z T ✓ z Yt ◆ ✓ x Xt ◆ d+1 K0 K dt fT(z/x)= ThT0 hT hT , z2R. 1 ZT ✓x Xt◆ d K dt ThT0 hT As usual, K0 and K are density functions on R and Rd, respectively, and hT ⌘ h(T ), T > 0, is a real positive function. More precisely, FT takes the form FT(y/x)= NT (x,y), with and 1 Z T ✓ y Yt ◆ ✓ x Xt ◆ NT (x,y)= DT(x)=dK dt, DT (x) K dt ThT0 hT hT d G 1 ZT ✓x Xt◆ ThT0 hT where G is the distribution function corresponding to K0. ]]>0,andG0 = (X0); • Ft = {(Xs,Ys):0s0,andF0 = (X0,Y0); • St, = {(Xs,Ys),Xr :0s 0, > 0, and S0, = {(X0,Y0),Xr :0r }, >0. In the sequel, E will denote the real interval [a, b], a < b. For easy reference, the assumptions needed to derive the announced results are gathered thereafter. (H1) Forevery t>0, >0, (i) the conditional density of Xt with respect to Gt , fGt , exists Xt and is lipschitzian on Rd (if t < 0, take Gt = G0); (ii) the conditional density of Xt with respect to Ft , fFt , exists Xt (iii) the conditional density of (Xt,Yt) with respect to Ft , fFt , (ift <0,takeFt =F0); exists; Z T (Xt ,Yt ) (ift <0,takeSt , =S0, ). (H3) Forallx2Rd,thereexistrealconstantsC>0, >0and >0such that, for all (y1, y2) 2 E2 and (x1, x2) 2 V(x) ⇥ V(x), we have |F(y1/x1) F(y2/x2)|C |y1 y2| +kx1 x2k , where V(x) denotes a neigbourhood of x and k · k the euclidean norm of Rd. (H4) The kernel K verifies: (i) K is lipschitzian and has compact support, k⇤ = sup |K(x)|; (iv) lim 1 fGt (x)dt=g(x)a.s., x2Rd. T!+1T0 Xt (H2) 8t 0, 8 >0, 8y2E, E✓G✓y Yt ◆.St , ◆=E✓G✓y Yt ◆.Xt◆ a.s. hT hT Z (ii) kxk K(x) dx < +1. x2Rd Rd ]]> 0, in order to prove Theorem 3.1 it is enough to establish the almost sure convergence to zero of Ai,T (x), for i = 1, 2, 3. (↵) We firstly analyse the behavior of A1,T (x). Note that A1,T (x) = 1 sup |CT (x, y)|, where DT (x) y2E CT (x,y)=NT (x,y) F(y/x)DT (x). Equivalently, CT (x, y) is given by 1 ZT ✓ ✓x Xt◆ ✓ ✓y Yt◆. ◆ . ◆ d E K E G St , F(y/x) Ft dt. ThT 0 hT hT DT (x) A3,T (x) = DT (x) DT (x) . Then, (H2) implies that 1ZT✓✓x X◆ ◆ CT (x,y)= d E K t (Xt)/Ft dt, ThT 0 hT with (Xt)=E✓G✓y Yt ◆/Xt◆ F(y/x). hT But, for every w 2 ZRd such that g(w) > 0, we have +1 (w) = K0 (u) (F (y hT u/w) F (y/x)) du 1 ]]> 1, 8>< b a b a > if 2N p = > h T h T T >b a b a : h +1if h 2/N TT where [u] denotes the integer part of the real number u. Dividing E = [a, b] into the intervals Ij,T =⇥a+(j 1)h T,a+jh T⇥, j2{1,...,pT 1}, and we may writeA2,T (x) IpT,T =⇥a+(pT 1)h T,b⇤, max sup NT (x, y) N T (x, y) j2{1,...,pT } y2Ij,T max NT (x,yj,T) NT (x,yj,T) , j2{1,...,pT } max sup NT (x,yj,T) NT (x,y) , j2{1,...,pT } y2Ij,T with A(1) (x)= 2,T A(2)(x)= 2,T A(3)(x)= 2,T 2,T 2,T 2,T max sup |NT (x,y) NT (x,yj,T)|, =  j2{1,...,pT } y2Ij,T A(1) (x) + A(2) (x) + A(3) (x), yj,T beinganarbitrarypointinIj,T,j2{1,...,pT}. The fact that G is absolutely continuous and (H5) (i) yield ! (1) A2,T (x)  max sup j2{1,...,pT } y2Ij,T = DT(x)O h 1 , T 1 d T hT Z T 0 K ✓ x Xt ◆ | y yj,T | hT hT k0 dt ]]> 0, ⇣ (2) ⌘ P A2,Tn(x)>✏   = pTn max P j2{1,...,pTn} 8(b a) h exp Tn Z ! Tn (1) 8(b a)h T C1✏2 Tn n Tn h2d Tn . Wt,Tn(x,yj,Tn)dt > ✏ 0 C1 ✏2 Tn h2d Tn log Tn Then, for T large enough, P ⇣A(2) (x) > ✏⌘ is bounded by C T C1 ✏2L1 , 2 n2d under assumption (H6), for some positive constants C2 and L1. n 2,Tn A suitable choice of and L1 assures that Tn 2d < +1, implying n=1 2,Tn ( ) To conclude the a.s. convergence to zero of A3,Tn(x) it su ces to apply Lemma. Lemma 4.2 with i = 0 and to use a similar reasoning. ⇤ We now state our main result. Theorem 3.2. Under the assumptions of Theorem 3.1, assume that f(·/x) is bounded. For all ↵ 2]0, 1[, we have |q↵,T (x) q↵ (x)| ! 0 a.s.. T !+1 Proof. Observe that, for each ↵ 2 ]0, 1[, we may choose a, b 2 R such that q↵(x) 2 [a,b]. Using this fact and the boundness of f(·/x), we get |q↵,T (x) q↵ (x)| = O( sup |FT (y/x) F (y/x)|) a.s., y2[a,b] by the argument presented in [7, Lemma 6.7]. 4. Auxiliary results ⇤ Lemma 4.1. Suppose that conditions (H1) (i), (iv), (H4) (ii), (H5) (ii) hold. If K has compact support, we have |DT (x) g(x)| ! 0 a.s. and DT (x) g(x) ! 0 a.s.. T !+1 T !+1 Proof. Please see [13, p. 78]. ⇤ P 2 +1 C1 ✏ L1 the a.s. convergence of A(2) (x) to zero, as n ! +1, via the Borel-Cantelli ]]> 0, Tn(i) ! 22d P Wt,Tn(x,y)dt >✏ 4 exp C1✏ Tn hTn 0 S with C1 a positive constant. n Proof. Fix i in {0, 1}. Writing [0, Tn] = [Tk 1, Tk], we get Z T n W (i) k=1 Xn (x, y) dt = Xn U (i) k,Tn V(i) (x,y)=Z Tl ⇣E⇣Z(i) (x,y).Fl 1, ⌘ E⇣Z(i) (x,y).Ft ⌘⌘dt. l 1,Tn t,Tn t,Tn (x, y) + with ZTk⇣ ⇣.⌘⌘ 0 t,Tn U(i) (x,y)= Z(i) (x,y) E Z(i) (x,y) Fk 1, dt, k,Tn t,Tn t,Tn Tk 1 Note that U(i) k,Tn k=1,...,n ` 1,Tn `=2,...,n respectively, verifying k=1 l=2 ⇣Tl 1 ⌘ ⇣ ⌘ V (i) l 1,Tn and V(i) (x,y) are martingale (x,y) di↵erences with respect to the fields (Fk, )k=1,...,n and (F` 1, )`=2,...,n, (x, y) and U(i) (x,y)  2k⇤ 1 , k , T n T n h dT n V (i) (x, y)  2 k⇤ 1 , ` 1,Tn Tn hdTn k = 1,...,n, ` = 2, . . . , n. ]]> ( ✏ 2 T n 2 h 2 d )  2exp Tn aswellas P =2exp Tn , 32 k⇤2 Xn `=2 ✏! (x, y) > ( ( ✏2Tnh2d) ) ⇤ V (i) ` 1,Tn ✏2 Tn2 h2d Tn k,Tn 2 k=1 32k⇤2n 2 ( ✏2Tnh2d)  2exp Tn . 2 exp 32 k⇤2 (n 1) 2 32 k⇤2 Then, taking C1 = 1 32 k⇤2 2 , we have the announced result. 5. Final remarks A straightforward improvement of Theorem 3.2 is the almost sure consis- tency of q↵,T(x), uniformly on x belonging to a compact subset of Rd. In line with the works of Didi and Louani [15] and Didi [13], we would deduce the almost sure convergence of the corresponding nonparametric predictor. In addition, it will be interesting to study consistency properties with rates of convergence. Acknowledgments We kindly acknowledge the support of the Department of Mathematics of the University of Coimbra. We are also grateful to the referee for his valuable suggestions and comments which greatly improved the text. References [1] G. Banon, Nonparametric identification for di↵usion processes, SIAM J. Control Optim. 16 (3), 380–395, 1978. [2] V. Bergelson, A. Leibman, and C. G. Moreira, From discrete to continuous-time ergodic theorems, Ergodic Theory Dynam. Systems 32 (2), 383–426, 2012. [3] D. Blanke and D. Bosq, Regression estimation and prediction in continuous time, J. Japan Statist. Soc. 38 (1), 15–26, 2008. [4] D. Bosq, Nonparametric statistics for stochastic processes, Springer-Verlag, New York, 1998. [5] D. Bosq and D. Blanke, Inference and prediction in large dimensions, John Wiley and Sons, Chichester, 2007. ]]> 0 the traditionally weakly informative priors for the PoINAR(1) ([45]) are chosen: a non-informative Beta prior with parameters a = 0.01,b = 0.01 and a a non-informative Gamma prior with parameters c = 0.01, d = 0.01, respectively. The prior specifications for pt, the probability of contamination is a Beta distribution with parameters (g = 5, h = 95), with expectation E(pt) = 0.05, reflecting the belief that outliers occur occasionally. The prior for the mean size of the outliers is a non informative Gamma distribution. The result of applying the outlier detection methodology is represented in Figure 3 indicating the occurrence of an outlier at time t = 224 with high probability. (a) 0 50 100 150 200 250 t (b) 0 50 100 150 200 t Figure 3. IP time series (a) and posterior probability of out- lier occurrence at each time (b). 0.0 0.4 0.8 02468 ]]>un,m , 8B✓]0,1]2. (1.2) nm In this paper we study the convergence in distribution of the sequence of point processes of exceedances, {Nn,m}, when the correlations of the random field {Xn,m} satisfy a strong dependence condition, specified by rn,m ln(nm) ! 0, n,m ! +1. (1.3) The case = 0 is studied in [2] for stationary gaussian random fields and in [6] for non stationary ones. In fact, considering that {Xn,m} is a non stationary gaussian random field, in [6] is studied its extremal behaviour, assuming that the correlations r(i1 ,i2 ),(j1 ,j2 ) satisfy |r(i1 ,i2 ),(j1 ,j2 ) | < ⇢|i1 j1 |,|i2 j2 | , for some sequence {⇢n,m} such that ⇢n,m ln(nm) ! 0, n,m ! +1, ⇢n,0 lnn ! 0, n ! +1and⇢0,mlnm!0, m!+1. We end this section recalling the underlying main results for gaussian se- quences of real random variables. Throughout this work and denote, respectively, the standard gaussian distribution function and its density. Consider that {Xn} is a stationary standard gaussian sequence, with cor- relations sequence {rn } satisfying rn log n ! 0, n ! +1. The sequence of point process of exceedances of the high level un by X1, X2, . . . , Xn is given by Nn(B)=#{j 1:j/n2B,Xj >un},foreachBorelsubsetBof]0,1].Itis well known that (cf. Leadbetter, Lindgren, and Rootz´en [5]), if un is such that n(1 (un)) ! e x, n ! +1, x 2 R, the sequence {Nn} converges in distribu- tion to a Cox process with stochastic intensity exp( x +p2 Y ), where Y is a standard gaussian random variable. If = 0, the limit in distribution of {Nn} is, obviously, a Poisson process with intensity = e x. As a trivial corollary, the sequence of maxima {Mn }, with Mn = max{X1 , . . . , Xn }, under the clas- sicalnormalizationun :=un(x)=x/p2lnn+p2lnn ln(4⇡lnn)/(2p2lnn), ]]> 0 and wi,j = max{|ri,j |, |vi,j |}. (i,j )2In,m |ri,j vi,j|exp u2n,m , 1+wi,j In order to specify an approximation for the distribution of the vector {Xi,j : (i, j ) 2 Qn,m } of the gaussian random field, in the next lemma we give conditions which imply that the upper bounds in (2.1) are asymptotically zero. This lemma is an extension of Lemma 5.2.1 of [2], where only the assump- tion rn,m ln(nm) ! 0, n, m ! +1, is considered. It should be remarked that suitable conditions on the behaviour of the sequences {r0,j}j and {ri,0}i are also needed. Lemma 2.2. Consider that {rn,m} and {un,m} are real sequences, is a non negative real number and ⇢n,m = /ln(n m). Assume that {n m(1 (un,m ))}n,m , {rn,0 ln n}n Xand {r0,m ln m}m are bounded sequ!ences and that (1.3) holds. Then nm |ri,j ⇢n,m|exp u2n,m !0, n,m!+1, (2.2) 1+wi,j with wi,j = max{|ri,j|,⇢n,m} and In,m defined in Lemma 2.1. (i,j )2In,m ]]> 0. When = 0, the proof of (2.2) for (i,j) in In,m\{(0, 1), (1, 0)} can be seen in [2]. By now we also assume that {un,m} is a sequence of normalized levels, that is nm(1 (un,m))!⌧, n,m!+1. Consider (k, `) = sup wi,j and let ↵ be a real number such that i k,j ` 0<↵< 1 (1,0)  1 (1,1). 1+ (1,0) 1+ (1,1) Take also p := [n↵], q = [m↵], := (1,1) and ⇤ := (1,0). We split the sum in (2.2) into eight terms. Namely, with A1 := {i 2 N : 1ip},A2 :={i2N:p0, and 2ln(nm) ⇠1, n,m!+1, u 2n , m ! +1. Considering un,m = x/an,m + Xp Xq u2n,m nm |ri,j ⇢n,m|exp 1+wi,j i=1 j=1  2n1+↵m1+↵ exp⇣ u2n,m ⌘ 1+ ⇣ ⇣u2 ⌘⌘2  2n1+↵m1+↵ exp n,m 1+ 2 Kn1+↵ 2 m1+↵ 2 (u )2 1+ 1+ n,m 1+ 1 1+ 1+ K (nm)1+↵ 2 (ln(nm)) 1 !0, n,m!+1, n, m bn,m, x 2 R, similarly, we obtain a = (2 ln nm)1/2 and b n,m n,m a 1 ln(4⇡ ln(n m))/2. (2.3) n,m n,m = a For the term concerning A1 ⇥ B1, we have s!uccessively ]]> 0, we obtain, for n and m large enough, ri,j ln(ij) ln(nm↵) ln(nm↵) ln(nm) 2 ⇥ i=p+1 j=1 ln(ij) " !#2 n2m u2 1+ (p,1) n2m ln p 2 ln p exp n,m  K lnn 2 lnn (ln(nm))k+ln p (nm) k+ln p K 1 n2 2↵lnn m1 2↵lnn (lnn+lnm) ↵lnn 1 k+↵ ln n k+↵ ln n k+↵ ln n lnn 1 k+↵lnn k+↵lnn Kn 2k (lnn) k (lnm)↵lnn m1 2 ↵lnn k+↵lnn k+↵lnn (2.5) exp⇣ 2klnn ⌘ ⇣k+↵lnn⌘(lnm) ↵lnn m1 2 ↵lnn =K K (lnm) ↵lnn m1 2 ↵lnn . 1 2 k+↵ ln n k+↵ ln n k+↵ ln n k+↵ ln n exp klnlnn k+↵lnn Furthermore, it can be shown that lnn Xn Xq ri,j  lnn Xn Xq 1 |ri,j ln(ij) | n i=p+1 j=1 ln(i j) n i=p+1 j=1 ln(i j)  1 m↵ sup |ri,j ln(i j) |. (2.6) k+↵lnn 1+↵ 2↵ ln n < 0, for n large, for the first sum in the right hand side of (2.4), k+↵lnn ↵ i 1,j 1 Thus, due to (2.5) and (2.6) and taking into account that ↵ ln n < 1 and ]]> 0, we get n m Xn Xm r i , j e x p u 2n , m ! i=p+1 j=q+1 ln(n m) 1 + wi,j !! 2 n2m2 u2 1+ (p,q) ln(n m) exp n,m ln(nm) 2 nm  K n2m2 ln(nm) k+↵2ln(nm) :i>p,j>q i,j ln(n m) i=p+1 j=q+1 nm n m ; ⇥ 8< Xn Xm r + :i=p+1j=q+1 i,j ln(ij) ln(ij) 9= ln(nm) ; ⇥ 8< s u p | r l n ( i j ) | + Xn Xm l n i j 1 9= ✓ ◆ ↵2 ln(n m) 1 ln(n m) (n m)2 deduce i 1 u2n,m ! nm Xp i=1 |ri,0 ⇢n,m|exp 1+wi,0 exp⇣ klnln(nm)⌘ ⇢ ⇣k+↵2 ln(nm)⌘⇥ o(1)+ Z 1Z 1 exp 2kln(nm) ln(nm) 0 0 |lnxy|dxdy K2 K3{o(1)+o(1)}=o(1), n,m!+1. k+↵2 ln(n m) Let us consider now the sums associated with A1 ⇥ {0} and A2 ⇥ {0}. From nowonwetake 0.Duetothefactthatsupwi,0  sup wi,j = ⇤,we i 1,j 0  2n1+↵m Kn1+↵ 2⇤m1 2⇤ (ln(nm)) 1⇤ 1+ K(nm)1+↵ 2 ⇤ ln(nm)!0, n,m!+1. !! 2 ⇤ u2 1+ exp n,m 2 1+ 1+ 1+ ]]>M,withMlargeenough, we get vn,m  un,m. Then, since (2.2) holds for {vn,m} it follows the same for {un,m }. ⇤ 3. Point processes of exceedances Consider the Cox process N defined over ]0,1]2, with stochastic intensity givenbyexp x +p2 Y ,wherex2R, 2R+0 andY isastandard gaussian random variable. The distribution of N is defined, in terms of its probability mass function, by P \q ! N(Bi) = si i=1 Z +1 Yq (Bi)e x +zp2 = s ! 1 i=1 i ! x +zp2 ⇥ exp( (Bi)e ) (z)dz (3.1) where, for q 1, s1,...,sq are non negative integers, B1,··· ,Bq are disjoint subsets of ]0, 1]2 and represents the Lebesgue measure. ]]>x)⇡P(✏21 >x). For a general BL(p, q, m, k), we have P(|X |>x)⇡P(✏k+1 >x). t1 Hence, while for linear models, Xt and ✏t are tail equivalent, for BL(p, q, m, k) models X is tail equivalent to ✏k+1 (see Resnick [31]). tt When innovations have lighter tails, for example Gaussian, then the relationship between the tails of input-output series is more compli- cated. However, typically the tail of the output series Xt is regularly varying when the innovations are Gaussian (Turkman and Amaral Turkman [40]). These results clearly indicate why and how bilinear processes produce extreme observations and show how useful they can be in modelling heavy-tailed dependent data. These features are evident in the simulated data sets that are plotted in Figure 1. When the innovation process is Gaussian (light-tailed), the simulated sample does not exhibit very large values (left panel). On the other hand, when Xt −5 0 5 Xt 0 50 100 150 200 250 300 350 Xt 0 10000 20000 30000 40000 50000 0 100 200 300 400 500 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 ttt Figure 1. Simulated bilinear data sets - N(0,1) innovations (left), Pareto(2.5) innovations (center) and Pareto(1.5) innovations (right). ]]> 0, with ↵ = 2.5 and ↵ = 1.5, the process can generate very large values (center and right panels). It is known that, for ↵ = 2.5, the Pareto distribution, with distribution function given by F (x) = 1 x ↵, x > 1 and ↵ > 0, has both finite mean and variance, but when ↵ = 1.5, although the mean is finite, the variance is not. The situation can be more extreme if ↵  1 because neither the mean nor the variance are finite. As mentioned before, the conditions that guarantee invertibility and sta- tionarity of bilinear process can only be derived for some particular models (see Subba Rao and Gabr [36]), such as the simple bilinear process which has only one bilinear term. Invertibility is a fundamental requirement if the model is to be used for prediction. Let Yt be a simple first order bilinear process, BL(1,0,1,1), given by Yt = aYt 1 + bYt 1✏t 1 + ✏t, (2.2) where {✏t} are iid rvs. A su cient condition for the process to be stationarity is given by Pham and Tran [30]: the parameters a, b and 2 must satisfy the condition a2 + b2 2 < 1, where |a| < 1. A su cient condition for invertibility is that the parameters a and b are such that b2E(Yt2)  1, as proved by Pham and Tran [30]. They also show that, provided the {et} are Gaussian rvs, the condition 2(1+a)b4 4 +2(1 a)b2 2 (1 a)2(1+a)0 (2.3) is su cient for the simple bilinear process to be invertible. Additional condi- tions on a, b and 2 are required if we want the bilinear process to have up to the fourth moment finite. In this case, besides | a |< 1 and a2 + b2 2 < 1, the following (necessary) conditions | a3 +3ab2 2 |< 1 and a4 +6a2b2 2 +3b4 4 < 1 have to be satisfied (Sesay and Subba Rao [34], Kim et al. [24], as referred by Leon-Gonzalez and Yang [25]). All these conditions severely restrict the parameter space while searching for admissible estimators for the parameters. Although bilinear processes do not satisfy the Markov property, it is possible to write them as state space models (SSM) satisfying a Markovian property, which may be very helpful in several aspects, namely for using SMC algo- rithms. If the observable process Yt is a BL(1,0,1,1) given in (2.2) then, solving iteratively for Yt, we get, after n iterations, n n 1 " j # Yt =Y(a+b✏t i)Yt n +X Y(a+b✏t i) ✏t j +✏t. i=1 j=1 i=1 If Xt = (a + b✏t)Yt, then Xt = (a + b✏t)Xt 1 + (a + b✏t)✏t ]]>