Introduction Random recurrence equations have been used in numerous elds of applied probability. We refer for instance to Kesten (973), Vervaat (979)

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1 Extremal behavior of the autoregressive process with ARCH() errors Milan Borkovec Abstract We investigate the extremal behavior of a special class of autoregressive processes with ARCH() errors given by the stochastic dierence equation q n = n + + n " n ; n N ; where ("n) nn are i.i.d. random variables. The extremes of such processes occur typically in clusters. We give an explicit formula for the extremal index and the probabilities for the length of a cluster. AMS 99 Subject Classications: primary: 60G70, 60J05 secondary: 60F05, 60G55 Keywords: ARCH model, autoregressive process, compound Poisson process, coupling, extremal behavior, extremal index, Frechet distribution, heavy tail, heteroskedastic homogeneous Markov process, recurrent Harris chain, separating sequence, strong mixing Center of Mathematical Sciences, Munich University of Technology, D-8090 Munich, Germany, borkovec@mathematik.tu-muenchen.de,

2 Introduction Random recurrence equations have been used in numerous elds of applied probability. We refer for instance to Kesten (973), Vervaat (979) and Embrechts and Goldie (994). Stochastic models in nance are an important eld of application for random recurrence equations. Over the last years a variety of these models have been suggested as appropriate models for nancial time series (see e.g. Priestley (988), Tong (990), Taylor (995)). Due to the random recurrence structure, many of these models possess the property that their conditional variance depends on the past information (conditional heteroskedasticity). Empirical work has conrmed that such models t quite many types of nancial data. The most known examples of volatility models in nance with random recurrence structure are autoregressive conditionally heteroskedastic processes (ARCH). These models were introduced by Engle (98). They serve as special exchange rate or asset price models and are very popular in econometrics. In a series of papers, the ARCH models have been analyzed and generalized, see for instance the survey article by Bollerslev, Chou and Kroner (99) and the statistical review paper by Shephard (996). The class of autoregressive (AR) models with ARCH errors proposed by Weiss (984) are a natural extension of ARCH processes. These models are also called SETAR-ARCH models (selfexciting autoregressive). They are dened by the random recurrence equation n = f( n ; :::; n k ) + n " n ; n k ; (.) where f is a linear function in its arguments, the innovations (" n ) nn are i.i.d. symmetric random variables with mean zero and n is given by n = 0 + p j= j n j ; 0 > 0; ; :::; p ; p > 0; (.) for some p. These models combine the advantages of AR models which target more on the conditional mean of n given the past and ARCH models which concentrate on the conditional variance of n (given the past). Autoregressive models with ARCH errors capture the structure of nancial data quite well, i.e. the tendency of volatility clustering and the fact that unconditional price and return distributions tend to have fatter tails than the normal distribution. Statistical and/or probabilistic properties of such models have been investigated by Weiss (984), Diebolt and Guegan (990), Maercker (997) and Borkovec and Kluppelberg (998). In the present paper we study the extremal behavior of AR processes with ARCH errors. We focus on the AR() process with ARCH() errors, i.e. f( n ; :::; n k ) = n for some

3 R and n is given in (.) with p =. This Markovian model is analytically tractable and serves as a prototype for the larger class of models (.). Furthermore, in the special case = 0 we get just the ARCH() model of Engle (98) and hence our results for the extremes will be an extension of the results in de Haan, Resnick, Rootzen and de Vries (989). Extremal behavior of a Markov process ( n ) nn behavior of the maxima M n = max kn k ; n : is for instance manifested in the asymptotic The limit behavior of M n is a well-studied problem in extreme value theory. Two review paper on this and related problems are Rootzen (988) and Perfekt (994). For a general overview of extremes of Markov processes, see also Leadbetter, Lindgren and Rootzen (983) and the references therein. Loosly speaking, under quite general mixing conditions, one can show that for n and x large where F is the stationary distribution function of ( n ) nn P (M n x) F n (x) ; (.3) and (0; ) is a constant called extremal index. A natural interpretation of is that of the reciprocal of mean cluster size (see e.g. Embrechts, Kluppelberg and Mikosch (997, Chapter 6) and the references therein). The practical implication of (.3) is that dependence in data does often not invalidate the application of classical extreme value theory. There are many methods for determing the extremal index. However, most are very technical and often useless in practice. An alternative is then to estimate from the data. For the AR() process with ARCH() errors we derive an explicit formula for the extremal index. We furthermore investigate the point process of exceedances of a high threshold u of ( n ) nn which characterizes the extremal behavior of the process in detail. This point process converges in distribution to a compound Poisson process with a well-specied intensity and a well-specied distribution of the size of the jumps. The paper is organized as follows: in section we present the model and introduce the required assumptions on the innovations (" n ) nn. The conditions are the same as in Borkovec and Kluppelberg (998), namely the so-called general conditions and the technical conditions (D:) (D:3). The general conditions guarantee the existence of a stationary version of ( n ) nn whereas (D:) (D:3) allow us to describe the tail behavior of the stationary distribution. We present furthermore some results on the AR() process with ARCH() errors ( n ) nn and on the 3

4 related process (Z n ) nn = (ln(n)) nn. It turns out that the process (Z n ) nn is crucial for the study of the extremal behavior of ( n ) nn. We show in Lemma.3 that (Z n ) nn behaves above a high threshold asymptotically as a random walk with negative drift which can be completely specied. Theorem. collects some known results on the AR() process with ARCH() errors ( n ) nn which were proved in Borkovec and Kluppelberg (998). In particular, the stationary distribution of ( n ) nn has a Pareto-like tail. Section 3 contains the main results (Theorem 3.) concerning the extremal behavior of ( n ) nn. We interprete these results and present some simulations. We conclude the paper in section 4 with the proof of Theorem 3.. Preliminaries We consider an autoregressive model of order with autoregressive conditional heteroskedastic errors of order (AR() model with ARCH() errors) which is dened by the stochastic dierence equation where (" n ) nn in addition the inequality q n = n + + " n n ; n N ; (.) are i.i.d. random variables, R; ; > 0 and the parameters and satisfy E(ln j + p "j) < 0 : (.) This condition is required to guarantee the existence and uniqueness of a stationary distribution. Let " be a generic random variable with the same distribution as " n. Throughout this paper, we assume the same conditions for " as in Borkovec and Kluppelberg (998). These are the so-called general conditions: " is symmetric with continuous Lebesgue density p(x) ; " has full support R ; (.3) the second moment of " exists : and the technical conditions (D:) (D:3): (D.) p(x) p(x 0 ) for any 0 x < x 0. 4

5 (D.) For any c 0 there exists a constant q = q(c) (0; ) and functions f + (c; ); f (c; ) with f + (c; x); f (c; x)! as x! such that for any x > 0 and t > x q p( x p + c + t ) p( x + t p ) f +(c; x) ; + t + t p( x p + c t ) p( x t p ) f (c; x) : + t + t (D.3) There exists a constant > 0 such that p(x) = o(x (N +++3q)=( q) ) ; as x! ; where N := inffu 0 ; E(j p "j u ) > g and q is the constant in (D:). There exists a wide class of distributions which satisfy these assumptions. Examples are the normal distribution, the Laplace distribution or the Students distribution. Conditions (D:) (D:3) are necessary for determing the tail of the stationary distribution. For further details concerning the conditions and examples we refer to Borkovec and Kluppelberg (998). Note that the process ( n ) nn is evidently a homogeneous Markov chain with state space R equipped with the Borel -algebra. The transition kernel density is given by P ( dy j 0 = x) = p p( y x p )dy ; (.4) + x + x The next theorem collects some results on ( n ) nn from Borkovec and Kluppelberg (998). Theorem. Consider the process ( n ) nn in (.) with (" n ) nn satisfying the general conditions (.3) and with parameters and satisfying (.). Then the following assertions hold: (a) Let be the normalized Lebesgue-measure () := ( \ [ M; M])=([ M; M]). Then ( n ) nn is an aperiodic positive -recurrent Harris chain with regeneration set [ M; M] for M large enough. In particular, there exists a constant C (0; ) such that for any Borel-measurable set B and x [ M; M] P ( B j 0 = x) C (B) : (.5) (b) ( n ) nn is geometric ergodic. In particular, ( n ) nn has a unique stationary distribution and satises the strong mixing condition with geometric rate of convergence. The stationary df is continuous and symmetric. 5

6 (c) Let F (x) = P ( > x); x 0; be the right tail of the stationary df and the conditions (D:) (D:3) are in addition fullled. Then F (x) c x ; x! ; (.6) where c = jj p E + + " E ( + p ")jj (.7) j + p "j ln j + p "j and is given as the unique positive solution to E(j + p "j ) = : (.8) Furthermore, the unique positive solution is less than two if and only if + E(" ) >. Remark. (a) Note that E(j + p "j ) is a function of ; and. It can be shown that for " N(0; ) and xed, the exponent is decreasing in jj. This means that the distribution of gets heavier tails when jj increases. In particular, the AR() process with ARCH() errors has for 6= 0 heavier tails than the ARCH() process (see also Table 3 in Borkovec and Kluppelberg (998)). (b) Theorem. is crucial for investigating the extremal behavior of ( n ) nn. The strong mixing property includes automatically that the sequence ( n ) nn satises the conditions D(u n ) and (u n ). The condition D(u n ) is a frequently used mixing condition due to Leadbetter et al. (983) whereas the slightly stronger condition (u n ) was introduced by Hsing (984). Loosly speaking, D(u n ) and (u n ) give the \degree of independence" of extremes situated far apart from each other. This property together with (.6) implies that the maximum of the process ( n ) nn belongs to the domain of attraction of a Frechet distribution. We will specify the normalizing constants of the maxima and the limit distribution in section 3. In order to study the extremal behavior of ( n ) nn and ( n) nn we dene the auxiliary process (Z n ) nn := (ln( n)) nn which is again a regenerative, strongly mixing process. Since ( n ) nn follows (.) the process (Z n ) nn Z n = Z n + ln satises the stochastic dierence equation ( + p e Z n + "n ) ; n N ; (.9) where (" n ) nn are i.i.d. random variables that satisfy the general conditions and (D:) (D:3), the constants are the same as in our old process ( n ) nn 6 and Z 0 equals ln( 0 ) a.s.. Note that

7 the process (Z n ) nn is independent of the sign of the parameter since " n is symmetric. Hence we may w.l.o.g. in the following assume that 0. We will see that (Z n ) nn can be bounded by two random walks (Sn l;a ) nn and (Sn u;a ) nn from below and above, respectively. This result is essential for the study of the extremal behavior of ( n ) nn. Via results for (Z n ) nn, we prove for instance that the regenerative process ( n ) nn has nite mean recurrence times which allow us to consider only the extremal behavior of the stationary process ( n ) nn. The process (Z n ) nn will be also important in the proof of Lemma 4.. For the construction of the two random walks (S l;a n ) nn and (S u;a n ) nn we need some more denitions. With the same notation as before, let A a := f! j p e a + p "(!) e a= p(a; ; ; ; ") := ln ( p + e a + ") ; q(a; ; ; ; ") := ln r(a; ; ; ; ") := ln p e a + + p g ; (.0) e a= p p e a= " ( + e a + ") f"<0g ; (.) p " e a ( + e a + ") f"<0g : Note that q(a; ; ; ; "); r(a; ; ; ; ")! 0 a.s. for a!. Now dene S l;a n := n j= U a j and S u;a n := n j= V a j ; n N ; (.) where U a j := Aa + p(a; ; ; ; " j ) + r(a; ; ; ; " j ) A c a \f" j <0g + ln( + p " j ) f"j 0g (.3) and V a j := p(a; ; ; ; " j ) + q(a; ; ; ; " j ) (.4) for some a 0. The following lemma shows that the random walks dened in (.)-(.4) are really upper and lower bounds for (Z n ) nn above a high level. Lemma.3 Let a be large enough, N a := inffj j Z j ag and Z 0 > a. Then Z 0 + S l;a k Z k Z 0 + S u;a k for any k N a a.s. (.5) 7

8 Proof. We prove only the lower bound. The proof of the upper bound is similar but easier. Let x a be arbitrary. If " 0 it is obvious that Consider now " < 0, then ( + p e x + ") ( + p ") : (.6) p p ( + e x + ") ( + e a + ") = ( ")p p e a + e x + (e a e x ) " e a " : (.7) Note that we have a non-trivial lower bound of ( + p e x + ") if and only if It is straightforward that (.8) is equivalent to " > ( + p e a + ") e a " > 0 : (.8) p e a + + p e a= or " < From (.6), (.7) and (.9), we obtain p 8>< + e x + "(!) >: p e a + p : (.9) e a= ( + p "(!)) ;! f" 0g p ( + e a + "(!)) e a "(!) ;! A c a \ f" < 0g(.0) 0 ;! A a Now take logarithms and use the additive structure (.9) of (Z n ) nn. Remark.4 (a) If a is large enough then S u;a n Proof. Note that and S l;a n E(V a ) = E (p(a; ; ; ; " ) + q(a; ; ; ; " )) = E ln are random walks with negative drift. ( + p e a + " ) + p e a= ( " ) f" <0g! E(ln( + p " ) ) < 0 ; as a! ; where we used the dominated convergence theorem and (.) in the last step. Hence for a large enough the statement follows. (b) Let (S n ) nn := Pn j= ln ( + p " j ) S l;a k nn. For a " we have P! S k and S u;a a:s: k! S k ; (.) 8

9 for any k N, i.e. both random walks converge at least in probability to the same random walk. Furthermore, sup S l;a k k d! sup k S k and sup S u;a a:s: k k Proof. The a.s. convergence of (S u;a n ) nn and sup k S u;a k! sup k S k : (.) is straightforward since p; q and r converge a.s.. Consider therefore the lower random walk (S l;a n ) nn. Note that for a " P (A a )! 0 and hence A c a \f"<0g P! f"<0g and Aa\f"<0g P! 0 : (.3) Furthermore, p(a; ; ; ; " ) + r(a; ; ; ; " ) a:s:! ln ( + p " ) ; (.4) and therefore (.) holds. Finally we note that E max(0; U a ) = E max 0; + E max p(a; ; ; ; " ) + r(a; ; ; ; " ) 0; ln( + p " ) f" 0g A c a \f" <0g! E max(0; ln( + p " ) ); as a! ; (.5) where we used (.3), (.4) and the dominated convergence theorem. By Borovkov (976), Theorem, p.53, (.) and (.5) we derive that sup S l;a k k d! sup k S k : Lemma.3 characterizes the behavior of the process (Z n ) nn above a high treshold a and hence also the behavior of (n) nn. This is the key to what follows: the process (S n ) nn will determine completely the extremal behavior of (n). Recall from Theorem. that ( n ) nn is Harris recurrent with regeneration set [ e a= ; e a= ] for a large enough. Thus there exists in particular a renewal point process T 0 ; T ; T ; ::: which describes the regenerative structure of ( n ) nn. 9

10 Figure : Simulated sample path of (Z n) nn with parameters = 0:6; = ; = 0:4 and starting point Z 0 = 50 (solid line) and the corresponding random walks (S l;a n ) nnand (S u;a ) nnwith a = 0 (dotted lines), respectively. Note that the random walks are hardly distinguishable from each other and (Z n) nn for n 47. Hence they are extremely good bounds above the level a = 0. If the process falls far below the level 0 they are still very close, but are no longer bounds for (Z n) nn. The picture also conrms our statement that the random walks have negative drift and converge to the same limit. n Corollary.5 The renewal point process (T n ) nn 0 which describes the regenerative structure of ( n ) nn is aperiodic and has nite mean recurrence times C 0 = T 0 and C = T T 0. Proof. The renewal process can be constructed in the following way (see e.g. Asmussen (989), Section VI.3 for some background on regenerative processes): Dene := inffk j k [ e a= ; e a= ]g = inffk j Z k ag = N a and i+ := inffk > i j Z k ag for i = ; ; 3; : : :. Since (Z n ) nn by the random walk with negative drift (S u;a n ) nn and is above the level a dominated sup E(max(0; Z ) j Z 0 = x) < ; (.6) x( ;a] it follows that ; ; 3 ; ::: are well dened and have nite expectations. Now let M := inffi j I i = g and M j+ := inffi > M j j I i = g for j = ; ; 3; ::: with P (I = ) = P (I = 0) = C and independent of ( n ) nn where C is the constant in (.7). Note that P (M j M j = i) = C( C) i for i; j = ; ; ::: and M 0 = 0 : (.7) 0

11 From Asmussen (989), p.5 and (.5), the renewal process (T n ) n0 is now given by T n := Mn+ + ; n 0 ; and hence, by (.7) E(C 0 ) = E(T 0 ) E( M +) const E(M + ) < : Similar calculation shows that E(C ) < as well. Since the transition density of (Z n ) nn positive and continuous it follows nally that C is aperiodic. is As a consequence of Corollary.5 we may suppose in the following that the process ( n ) nn is stationary. One can show by a coupling argument that for any probability measure and any sequence (u n ) nn P max k u n P max k u n! 0; as n! ; kn kn where P denotes the probability law for ( n ) nn when 0 starts with distribution and is the stationary distribution. For the coupling argument one needs explicitly that the process ( n ) nn is regenerative and that the embedded renewal process is aperiodic and has nite mean recurrence time. We refer to Lindvall (99, Chapter II and III) for further details. 3 Extremal behavior of the AR() process with ARCH() errors In this section we present the main results concerning the extremal behavior of the AR() process with ARCH() errors and the accompanying squared process. Let ( b n ) nn be the associated independent process of ( n ) nn, i.e. b ; b ; ::: are i.i.d. random variables with the stationary distribution function of ( n ) nn. From (.6) and classical extreme value theory we obtain lim P n! (n = max kn b k x) = exp( c x ) ; x 0 ; (3.) hence the maximum of the associated independent process ( b n ) nn belongs to the domain of attraction of a Frechet distribution. In the dependent case we prove a similar result. The limit distribution is still a Frechet distribution but a constant occurs in the exponent. is called the extremal index of the process ( n ) nn and is a measure of local dependence amongst the exceedances over a high threshold by the process ( n ) nn. It has a natural interpretation as

12 the reciprocal of the mean cluster size. In order to describe the extremes in more detail, we also consider the point process (N n ) nn given by of exceedances of an appropriately chosen high threshold u n N n () := #fk=n j k > u n ; k f; :::; ng g (3.) and show that this point process converges to a compound Poisson process N. We derive the intensity and the distribution of the jumps which we denote by ( k ) kn. Note that in the extreme value theory for strong mixing processes the jumps equal the lengths of clusters of exceedances. For further background we refer to Leadbetter et al. (983), Rootzen (988) or Embrechts et al. (997, Section 8.). For the ARCH() process it was convenient to investigate rst the squared process. This is not the case for our model since we have a completely dierent structure due to the autoregressive part of ( n ) nn. Nevertheless, only for the squared process ( n) nn a comparison with results in the ARCH() case (see de Haan et al. (989)) is possible. The following theorem collects our results. Theorem 3. (a) Suppose ( n ) nn is given by equation (.) with (" n ) nn satisfying the general conditions (.3) and (D:) (D:3) with parameters and satisfying (.) and 0. Then lim P (n = max j x) = exp( cx ) ; x 0 ; (3.3) n! jn where P denotes the law for ( n ) nn when 0 starts with the distribution, solves the equation E(j + "j ) =, c is dened by (.7) and = Z P (sup k ky i= ( + p " i ) y )y dy : For x R, let N n be the point process of exceedances of the threshold u n = n = x by ; :::; n given by (3.). Then N n d! N; n! ; where N is a compound Poisson process with intensity cx and cluster probabilities k = k k+ ; k N ; (3.4) where k = Z P (#fj j jy i= ( + p " i ) > y g = k )y dy ; k N :

13 In particular, =. (b) Let ( n ) nn be the AR()-process with ARCH()-errors in (a) and ( n) nn the squared process. Then lim P (n = max n! j x) = jn exp( c() x = ) ; x 0 ; (3.5) where ; c are the same constants as in (a) and () = Z P (sup ky k i= ( + p " i ) y )y dy : For x R, let N () n be the point process of exceedances of the threshold u n = n = x by ; :::; n. Then N () n d! N () ; n! ; where N () is a compound Poisson process with intensity c () x = and cluster probabilities () k = () k () k+ ; k () N ; (3.6) where () k = Z P (#fj j jy i= ( + p " i ) > y g = k )y dy ; k N : In particular, () = (). Remark 3. (a) Theorem 3. is a generalization of the result of de Haan et al. (989) in the ARCH() case (i.e. = 0). They use a dierent approach which does not extend to the general case because of the autoregressive part of ( n ) nn. (b) Note that for the squared process one can describe the extremal index and the cluster probabilities by the random walk (S n ) nn, namely () k = Z 0 P (#fj j S j > xg = k ) e x dx ; k N : The description of the extremal behavior of ( n) nn by the random walk (S n ) nn is to be expected since by Lemma.3 and Remark.4 the process (Z n ) nn = (ln( n)) nn behaves above a high threshold asymptotically like (S n ) nn. Unfortunately, this link fails for ( n ) nn. Another possibility for proving statement (b) is to follow the work of Hooghiemstra and Meester 3

14 (995) using the regenerative structure of (Z n ) nn, Lemma.3, Corollary.5 and Remark.4(b). (c) Analogous to de Haan et al. (989) we may construct \estimators" for the extremal indices () and () k of ( n) nn, respectively, by b () = N N i= fsupjm S (i) j E (i) g and b () k = N N i= P f mj= ; for k (i) fs j > E (i) g=k g N ; where N denotes the number of simulated sample paths of (S n ) nn, E (i) are i.i.d. exponential random variables with intensity and m is chosen large enough. These estimators can be studied as in the case = 0 and " N(0; ) in de Haan et al. (989). In particular, () b () ( () ( () )=N) = is approximately N(0; ) distributed. Because of Remark 3.(b) this approach is not possible for ( n ) nn. We choose as \estimators" for and k for ( n ) nn b = N N i= Q j p fsupjm l= (+ "l )=P (i) g (3.7) and b k = N N i= f P mj= f Q j l= (+p " l )>=P (i) g=k g ; for k N ; (3.8) Q n where N denotes the number of simulated paths of ( ( + p l= " l )) nn, P (i) are i.i.d. Paretodistributed random variables with intensity, i.e. with distribution function G(x) = x, x 0, and m is large enough. These are suggestive estimators since Q n l= ( + p " l )! 0 a.s. as n! because of assumption (.). (d) Note that the extremal index of ( n ) nn is not symmetric in the parameter (see Table ). This observation is intuitively obvious since for > 0 the clustering is stronger by the autoregressive part than for < 0. 4

15 Threshold theta (67 blocks of size 60) K Figure : Estimated extremal index of a simulated sample path of ( n) 0n0000 with parameters = 0:8; = ; = 0:6 and " N(0; ) using the blocks method for the data (see Embrechts et al. (997), Section 8.). The length of a block is chosen as 60. The solid line is the numerically computed extremal index using (3.7), see also Table : : : : : : : : : : : Table : \Estimated" extremal index of ( n) nn in the case " N(0; ). We chose N = m = 000. Note that the extremal index decreases as jj increases and that we have no symmetry in. 5

16 jj : : : : : : Table : \Estimated" extremal index () of (n) nn dependent on jj and in the case " N(0; ). We chose N = m = 000. Note that the extremal index decreases as jj increases. jj Table 3: \Estimated" extremal index () and cluster probabilities ( k ) k6 of (n) nn dependent on and in the case " N(0; ). We chose N = m =

17 Table 4: \Estimated" extremal index and cluster probabilities ( k ) k6 of ( n) nn dependent on and in the case " N(0; ). We chose N = m = 000. Note that the extremal index for > 0 is much larger than for < 0. 7

18 Figure 3: Simulated sample path of ( n) nnwith parameters = 0:8; = ; = 0: (top, left), of (n) nnwith the same parameters (top, right), of ( n) nnwith parameters = 0:8; = ; = 0: (middle, left), of (n) nn with the same parameters (middle, right), of ( n) nn with parameters = 0; = ; = 0: (bottom, left) and of (n) nn with the same parameters (bottom,right) in the case " N(0; ). All simulations are based on the same simulated noise sequence (" n) nn. 8

19 4 The proof of Theorem 3. The proof of Theorem 3. will be an application of results in Perfekt (994) (see also the Appendix). In order to apply these results we need to check the assumptions in Theorem A. and A.. The next lemma provides a technical property for the squared AR() process with ARCH() errors (n) nn. It is the most restrictive assumption in Perfekt (994). Lemma 4. Let (p n ) nn be an increasing sequence such that p n n! 0 and n( p p n ) p n! 0 as n! ; (4.) where is the mixing function of ( n ) nn. Then for u n = n = x lim p! lim sup P ( max j > u n j > u 0 n) = 0 : (4.) n! pjp n Remark 4. (a) The strong mixing condition is a property of the underlying eld of a process. Hence is also the mixing function of ( n) nn and (Z n ) nn and we may work in all these cases with the same sequence (p n ) nn. (b) In the case of a strong mixing process, conditions (4.) are sucient to guarantee that (p n ) nn is a (u n )-separating sequence. The notion of a (u n ) separating sequence was rst introduced by O'Brian (989) and describes somehow the interval length needed to accomplish asymptotic independence of extremal events over a high level u n in separate intervals. For a denition see also Perfekt (994). Note that (p n ) nn is in the case of a strong mixing process independent of (u n ). Proof. Note that P ( max pjp n j > u n j 0 > u n) = P (N a < p; max pjp n j > u n j 0 > u n) + P (p N a < p n ; max pjp n j > u n j 0 > u n ) + P (N a p n ; max pjp n j > u n j 0 > u n ) =: I + I + I 3 ; (4.3) where N a = inffj j Z j ag = inffj j j ea g as in Lemma.3. In order to get upper bounds of I ; I and I 3 we show rst that there exist constants C > 0 and N N such that for any n > N; x [e n ; e a ] and k N n P ( k > u n j 0 = x) C : (4.4) 9

20 Assume that (4.4) does not hold. Choose C; N > 0 arbitrary and > 0 small. Because of the continuity of the transition probability (i.e. equicontinuity on compact sets), there exist n > N; x [e n ; e a ]; k N and = () > 0 such that for any y (x ; x + ) \ [e n ; e a ] n P (k > u n j 0 = y) > C : (4.5) Let F denote the stationary df of ( n) nn. By Theorem. we have that lim n! n F (u n) = c x = ; (4.6) where c is given by the formula in (.7) and is the solution of (.8). Furthermore, by (4.5) we have n F (u n ) = Z Z ( ;) n P (k > u n j 0 = y)df (y) n P (k > u n j 0 = y)df (y) (x ;x+)\[e n ;e a ] > (C ) P ( 0 (x ; x + ) \ [e n ; e a ]) (C ) D ; where D := inf z[0;e a ] (F (z + ) F (z)) > 0 because F arbitrary this is a contradiction to (4.6). Now we estimate (4.3). I = = p l= p P N a = l; p n l= j=l+ p p n l= j=l+ p p n max j > u n pjp n > u 0 n P N a = l; j > u n 0 > u n E E l= j=l+ p p n + Furthermore, by (4.4), J l= j=l+ =: J + J : p p n l= j=l+ fna=lg P ( j > u n j l ) 0 > u n fna=lg f l e n g P ( j > u n j l ) E fna=lg f l <e n g P ( j > u n j l ) n E fna=lg f l e n g n P ( j > u n j l ) 0 is continuous. Since C > 0 is 0 > u n 0 > u n 0 > u n (4.7)

21 p p n l= j=l+ p n j= C p n n C n E fna=lg f l e n g > u 0 n C n P (N a < p j 0 > u n ) (4.8)! 0 ; as n! ; since p n = o(n). Similarly, with B l := f > ea ; :::; l > ea g for any l = ; 3; 4; ::: and B =, we obtain J = = = = p p n l= j=l+ p p n l= j=l+ p p n l= j=l+ p p n l= j=l+ p + E fna=lg f l <e n g E 0 > u n Bl P (l < e n jl ) > u 0 n q E Bl P ( l + + " l l) < e n l 0 E p n l= j=l+ p p n E l= j=l+ p p n + l= Bl \f l >0g P e n= = l q E E const p p n e n= a=! 0 ; as n! ; = l + Bl \f l <0g P e n= = l + q = l + 0 > u n < " l < e n= = l q= + 0 > u n A l < " l < e n= = l + q= + 0 > u n A l e n= a= Bl \f l >0g P p < " l < e n= a=! p > u 0 n e n= a= + Bl \f l <0g P p < " l < e n= a= + p > u 0 n! and therefore with (4.8) I! 0 as n!. Now we estimate lim sup n! I 3. Note rst that by the Markov inequality P max S u;a pjp j n = p n j=p Y j P > z m= p n j=p P e 4 Su;a j > e z 4 ( + p e a + " m ) p e a= " m f"m<0g =4 > e 4 z

22 p n e 4 z E j=p p n e 4 z j ; j=p where < such that E p p! j =4 ( + e a + " ) e a= " f" <0g (4.9) p ( + e a + " ) p =4 e a= " f" <0g for a large enough. This is possible because of (.) which implies that E(j+ p " j u ) < for all u (0; ) and the fact that p p =4 E ( + exp( a) + " ) e a= " f" <0g! E j + p " j = ; a! by the dominated convergence theorem. Thus from Theorem., Lemma.3, (4.9) and a large enough, lim sup n! Finally, note that I 3 lim sup P (N a p n ; n! P ( max lim sup n! = lim sup n! j=p Z 0 max Z 0 + S u;a pjp j > ln u n j Z 0 > ln u n ) n Z 0 + S u;a pjp j > ln u n j Z 0 > ln u n ) n P ( max S u;a pjp j > z) n e z dz (4.0) j = p : I P (p N a < p n ; max j > u n j > u 0 n) + P (p N a < p n ; max j > u n j > u 0 n) N a<jp n pjn a =: K + K : Similarly as for I and I 3, respectively, we derive that lim sup n! K = 0 and lim sup K n! = p : Now plugging all together and letting p! the statement follows. Corollary 4.3 Let (p n ) nn be the same sequence as in Lemma 4.. Then (p n ) nn is also a (u n ) separating sequence for ( n ) nn, where u n = n = x and x R arbitrary and lim p! lim sup P ( max j > u n j 0 > u n ) = 0 : (4.) n! pjp n

23 Proof. Because of Remark 4.(a) and (b), it is straightforward that (p n ) nn is a (u n ) separating sequence for ( n ) nn. Note furthermore that P ( max pjp n j > u n j 0 > u n) = P (max pjp n j > u n; 0 > u n) P ( 0 > u n) P (max pjp n j > u n ; 0 > u n ) P ( 0 > u n ) + P ( 0 < u n ) = P ( max pjp n j > u n j 0 > u n ) and hence the statement follows using Lemma 4.. Now we are nally able to prove Theorem 3.. Proof of Theorem 3.. The proof is an application of Theorem A.. We prove only statement (a), statement (b) follows along the same lines using Theorem A.. As stated already we may assume w.l.o.g. that ( n ) nn and P ( 0 > u + lim u x) = u! P ( 0 > u) is stationary. Let x R be arbitrary. Note that 8 < : ; + x 0 ( + x) ; + x > 0 lim P ( u! u x j 0 = u) = P ( + p " x) : By Corollary 4.3 and the strong mixing property of ( n ) nn all assumptions of Theorem A. are fullled and we have that the extremal index is given by = = Z Z P (#fj j ( P (max j ( jy i= jy i= ( + p " i ))Y 0 > g = 0 j Y 0 = y) y dy ( + p " i ) y ) y dy : The cluster probabilities can be determined in the same way and hence the statement follows. A Appendix The theorem below gives the extremal properties of a fairly large class of stationary Markov chains. The original version can be found in Perfekt (994, Theorem 3., p. 538). We present a simplied version of Perfekt's result which can be directly applied to our situation. 3

24 Theorem A. Suppose ( n ) nn is a stationary Markov chain which satises for some ( ; ) the following properties (i) F (u + g(u)x) lim u"x F F (u) = ( x) = + ; x ( ; ); where F is the stationary df, x F := supfx ; F (x) < g, y + := maxf0; yg and x F = and g(u) = u if < 0 x F < and g(u) = (x F u) if > 0 If = 0, then the auxiliary function g is unique up to asymptotic equivalence and strictly positive on (x 0 ; x F ) for some x 0 < x F. (ii) lim P ( ( u) ) = + x j 0 = u = H(x) u!x F g(u) for some df H on [0; ). Let furthermore (A n ) nn be an i.i.d. sequence with marginal df H and let Y 0 be a random variable independent of (A n ) nn. Dene the tail chain (Y n ) nn by Y n = A n Y n for n and denote by P the law of (Y n ) nn when Y 0 has distribution. Assume (dx) = x dx; x > and let (u n ()) be a sequence which satises lim n! n( F (u n()))! : (a) Assume D(u n ()) holds for each > 0. If for some 0 there is a D(u n ( 0 ))-separating sequence (p n ) nn such that lim p! lim sup P ( max j > u n j 0 > u n ) = 0 n! pjp n (A.) holds with u n = u n () then ( n ) nn has extremal index given by = P (#fn j Y n > g = 0) : 4

25 (b) Suppose ( n ) nn has extremal index > 0 and, for some > 0 satises (u n ( )) for each > 0. Suppose further there is a (u n ( ))-separating sequence (p n ) nn such that (A.) holds with u n = u n ( ). Then, for each > 0, N n := #fk f; :::; ng j k=n ; k > u n ( )g converges in distribution to a compound Poisson process N with intensity jump probabilities i given by i = P (#fn j Y n > g = i ) P (#fn j Y n > g = i) ; i N : The next theorem is an extension of Theorem A.. In some cases it is easier to apply then the last one. and Theorem A. (Extension of Theorem 3. of Perfekt (994), p. 543) Suppose ( n ) nn stationary Markov chain which satises is a F (u + g(u)x) lim = ( x) = + ; x ( ; ); u"x F F (u) where F is the stationary df, x F := supfx ; F (x) < g =, y + := maxf0; yg and g(u) = u for some < 0 : Suppose furthermore that inffx ; F (x) > 0g = and that lim P ( u! u x j 0 = u) = H(x) ; for some df H on ( ; ). Let (A n ) nn be an i.i.d. sequence with distribution H and dene the tail chain through Y n = A n Y n ; n, Y 0 being independent of (A n ) nn. Then, if ( n ) nn satises the conditions in (a) and (b) of Theorem A., the result of the theorem holds with the initial distribution given by (dx) := jj x = dx; for x >. Acknowledgement I wish to express my gratitude to Claudia Kluppelberg for her constant support and advice. I am also grateful to Alex McNeil for an Splus program which estimates the extremal index by the blocks method. References [] Asmussen, S. (987) Applied Probability and Queues. Wiley, Chichester. 5

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