Long strange segments in a long-range-dependent moving average

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1 Stochastic Processes and their Applications 93 (200) Long strange segments in a long-range-dependent moving average Svetlozar T. Rachev a, Gennady Samorodnitsky b; a Institute of Statistics and Mathematical Economics, Kollegium am Schloss, University of Karlsruhe, Postfach 6980, D-7628, Karlsruhe, Germany b School of Operations Research and Industrial Engineering and Department of Statistical Science, Cornell University, Ithaca, NY 4853, USA Received 30 May 2000; received in revised form 28 October 2000; accepted 30 October 2000 Abstract We establish the rate of growth of the length of long strange intervals in an innite moving average process whose coecients are regularly varying at innity. We compute the iting distribution of the appropriately normalized length of such intervals. The rate of growth of the length of long strange intervals turns out to change dramatically once the exponent of regular variation of the coecients becomes smaller than, and then the rate of growth is determined both by the exponent of regular variation of the coecients and by the heaviness of the tail distribution of the noise variables. c 200 Elsevier Science B.V. All rights reserved. MSC: Primary 60G0; 60F5; secondary 60G70 Keywords: Long-range dependence; Moving average process; Large deviations; Heavy tails; Regular variation; Extreme value distribution; Applications in nance; Insurance; Telecommunications. Introduction Given an ergodic stationary stochastic process X =(X ;X 2 ;:::) with a nite mean = EX and, one denes for every n =; 2;::: { R n (; X) = j i: 06i j6n; X } i+ + + X j ; (.) j i This research was done with port of SEW-EURODRIVE-Foundation (SEW-Group) that funded a visit of G. Samorodnitsky to University of Karlsruhe in summer of 999. Samorodnitsky s research was also partially ported by NSF grants DMS and DMI as well as by NSA grant MDA at Cornell University. Corresponding author. addresses: zari.rachev@wiwi.uni-karlsruhe.de (S.T. Rachev), gennady@orie.cornell.edu (G. Samorodnitsky) /0/$ - see front matter c 200 Elsevier Science B.V. All rights reserved. PII: S (00)

2 20 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 (dened to be equal to zero if the remum is taken over the empty set). We think of R n (; X) as the length of the longest time interval up to time n where the process seems not to do what the law of the large numbers tells it to do. Hence, a long strange interval. Of course, the numbers R n (; X) can be dened for any stochastic process, but they become really interesting in the ergodic stationary case. Awareness that such intervals can be fairly long is important in various applications in manufacturing, nance, insurance, genetics and computer science, see, e.g. Dembo and Zeitouni (993) and Manseld et al. (999). The rate of growth of the length of long strange intervals is related to both heaviness of the tail of the marginal distribution of the process and the dependence structure of the process. In fact, it has been suggested in Manseld et al. (999) that a fruitful way of thinking about the dichotomy between short and long range dependence in a family of ergodic stationary stochastic processes is via a possible change in the rate at which the length of long strange intervals grows with the sample size n: in one part of the parameter space the rate of growth is the same as in the iid case, and one says that parameters in this range lead to short-range-dependent models. In another part of the parameter space the rate of growth is higher; one calls it the long-range-dependent case. In this sense, crossing the boundary between the two parts of the parameter space is a phase transition. In the present paper we investigate the nature of such phase transition in the behavior of long strange interval for innite moving average processes. That is, we consider stochastic processes of the form X n = + n j Z j ; n=; 2;:::; (.2) where :::;Z ;Z 0 ;Z ;::: is a sequence of zero mean iid random variables, and is a constant. We assume that Z = Z 0 has balanced regular varying tails: P( Z )=L () ; P(Z ) P( Z ) = p; P(Z ) P( Z ) = q; (.3) as, for some and 0 p= q6. Here L is a slowly varying function at innity. We also assume that the coecients ( j ) are regularly varying and balanced. That is, there is a function : [0; ) [0; ) such that (t)=l 2 (t)t h (.4) as t and such that j j (j) = +; j j (j) = ; (.5) for some + ; 0, at least one of which is positive. Here { h max ; } 2 (.6)

3 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) and L 2 is a slowly varying function. It is well known that under the above conditions X ;X 2 ;::: is an ergodic stationary process whose marginal distribution satises P(X ) j [pi { j 0} + qi { j 0}]P( Z ) j [pi { j 0} + qi { j 0}]L () (.7) as. As it is usual, we use the notation X for a generic random variable with the same distribution as X. See, for example, Lemma A3:7 in Mikosch and Samorodnitsky (2000). Further facts on linear processes and regular variation can be found in Brockwell and Davis (99) and Resnick (987). An encyclopedic treatment of regular variation is in Bingham et al. (987). It turns out that for such innite moving average processes the phase transition mentioned above occurs as one crosses the boundary h =. In order to describe what happens, let us introduce some notation. For a regularly varying at innity function g with a positive exponent of regular variation, let g (y) = inf {s: g(s) y} (.8) be its generalized inverse. Then g is a nondecreasing regularly varying function with exponent = of regular variation. See Theorem :5:2 in Bingham et al. (987). We list additional properties of generalized inverses of regular varying functions in Lemma 3. below. Denoting by F the distribution function of the random variable Z we dene, for every n, ( ) a n = (n): (.9) F We immediately see from (.3) that the sequence (a n ) is regularly varying with exponent =. It has been established in Manseld et al. (999) that in the case h, for every P(a n R n (; X)6x) exp{ ( ) x (pm + ( ) + qm ( ) )} (.0) as n for every x 0. Here ( k ) + + M + ( ) = max j ; j k k and M ( ) = max k j=k ( k ) j ; k j j=k ; with the notation a + = max(a; 0) and a =( a) + for a real a. In fact, (.0) holds whenever the coecients ( j ) are absolutely summable. Our goal in this paper is to show that in the case h the normalization by a n as in (.0) is no longer sucient, and that the correct normalization depends both

4 22 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 on the tail index and the exponent of regular variation h. In fact, in the case h the length R n (; X) of the long strange segments grows at the rate faster than a n, which is regularly varying with exponent =. We regard this as an indication of long-range-dependence. Let a n be, once again, as in (.0). With the function in (.4), set b n = ( ) (a n ); (.) n. Observe that the sequence (b n ) is regularly varying with exponent =(h). Since h we see that a n =o(b n ) as n : It turns out that b n is the proper normalization of R n (; X) in the long-range-dependent case. That is, the main result of this paper stated and proved in the next section shows that b n R n (; X) converges weakly to a nondegenerate it. Roughly speaking, this means that R n (; X) grows as n =(h) in the case h and as n = in the case h. If one views the exponent h as a measure of the length of dependence, then an informal estimate of h can be obtained by plotting R n (; X) against the sample size n in the logarithmic scale. This approach is similar in its informality to R=S statistic used to estimate Hurst parameter, see Hurst (95), or Beran (994) for more information. However, while Hurst parameter is related to the behavior of a somewhat obscure functional of the data, the exponent h in our case is directly related to the length R n (; X) of the long strange segments, which are of obvious interest on their own. We note that the boundary case h = is more complicated, and the rate of growth of R n (; X) in this case is likely to depend on the interplay between the slowly varying factors in the regularly varying functions involved. It may be possible to use the rate of growth of the length of the long strange intervals to construct a test for long-range dependence. Moreover, it is very likely that similar behavior can be observed in many other families of stationary ergodic processes, and not only in the case of linear processes. How this plays out in practice is still under investigation. One has to realize, further, that before constructing a practical statistical procedure based on our iting results one will have to address many additional questions. This includes understanding the speed of convergence in the it theorems to the extent that it aects nite sample distributions. This also includes the question of selecting intelligently the parameter. This paper is organized as follows. In the next section we state and prove the main result. A part of the argument is organized as lemmas in Section 3. Numerical results demonstrating some of the practical issues related to our iting procedure are presented in Section Rate of growth of long strange intervals The main result of this paper is presented in the following theorem.

5 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) Theorem 2.. Let X =(X ;X 2 ;:::) be a stationary innite moving average process (:2) with noise variables satisfying (:3); and regularly varying coecients satisfying (:5) and (:6) with h. Assume that the function is; eventually; positive. Let b n be dened by (:). Then for every x 0 and P(b n R n (; X)6x) exp{ p( h) ( =h + + =h ) h ( ) x h } (2.) as n. Several remarks are in order. Remark 2.2. It follows from Theorem 2. that properly normalized length R n (; X) of the long strange segments has the so called h extreme value distribution as its weak it. See Section :2 in Resnick (987). Remark 2.3. Regular variation of the coecients guarantees that they are, eventually, of the same sign. We have assumed that the function is eventually positive. The situation is very similar in the case when it is eventually negative, except that the role of the left and right tails of the noise variables change. That is, (2.) remains true but the right tail weight p has to be replaced by the left tail weight q. Remark 2.4. Needless to say, a result similar to that in Theorem 2. holds also for the length of the intervals where the sample mean is too low. If one denes { R n (; X) = j i: 06i j6n; X } i+ + + X j j i for a, then the corresponding result for R n (; X) can be obtained by multiplying the entire process by and restating Theorem 2. in the language appropriate for the new process. The idea underlying the result in Theorem 2. is similar to the idea in the corresponding result of Manseld et al. (999) in the short-range-dependent case. Specically, it is very likely that most of the length R n (; X) is due to a single noise variable. Since X i+ + + X i+k = k + Z j (2.2) for every i and k, the above statement means that, as n, we have R n (; X)6x) P for all j = :::; ; 0; ;::: Z j 6k( ) P(b n for all xb n 6k6n and all i =0;:::;n k := P M (x;; n) (2.3) in the sense that the probability in the right-hand side in (2.3) is the main term in the probability in its left-hand side. The proof of Theorem 2. consists, basically,

6 24 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 of evaluating the behavior of the probability P M (x;; n) in the right-hand side in (2.3) and checking that the other terms in the probability in the left-hand side of (2.3) are negligible. Of course, the technical details are quite dierent between the short-range-dependent and long-range-dependent cases. Nonetheless, we do achieve some savings here by utilizing in the sequel available results from Manseld et al. (999). Now we are ready to prove Theorem 2.. Proof. For notational simplicity we may and do assume that = 0 and 0. We rst prove the theorem in the case + ; 0 in (.5). This implies, in particular, that j 0 for all j large enough (say, with j j 0, for some j 0 0). We claim that it is enough to prove the theorem in the case j 0 for all j: (2.4) Indeed, pose that we have proved the theorem under the assumption (2.4). With j 0 as just dened, we set () j = j ( j j 0 ); (2) j = j ( j j 0 ) for j and dene two innite moving averages X () and X (2) by X () n = () n j Z j; X (2) n = (2) n j Z j; n =; 2;:::. Observe that X n = X n () + X n (2) for all n, that the coecients ( () j ) are nonnegative and satisfy the same assumptions (.5) and (.6) that the coecients ( j ) do. Finally, only nitely many of the coecients ( (2) j ) are dierent from zero. Choosing an (0;), we have by Lemma 3.2 b n R n (; X) 6 b n max(r n ( ; X () );R n (; X (2) )) 6 b n R n ( ; X () )+b n R n (; X (2) ): However, (2.) is assumed to have been proved in the case of nonnegative coecients, (and the process X () has nonnegative coecients), and (.0) holds in the case of absolutely summable coecients (and the process X (2) has absolutely summable coef- cients). Therefore, for every x 0 inf P(b n R n (; X)6x) exp{ p( h) ( =h + + =h ) h ( ) x h }; and letting, 0 we conclude that inf P(b n R n (; X)6x) exp{ p( h) ( =h + + =h ) h x h }: (2.5) Similarly, for every 0 we have by Lemma 3.2 b n R n ( + ; X () )6b n max(r n (; X);R n (; X (2) )); so that b n R n (; X) b n R n ( + ; X () ) b n R n (; X (2) ):

7 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) A similar application of (2.) for the process X () and of (.0) for the process X (2), with a subsequent letting 0 gives us P(bn R n (; X)6x)6exp{ p( h) ( =h + + =h ) h x h } (2.6) for any x 0. Now (2.) follows from (2.5) and (2.6). Therefore, it is enough to prove the theorem under the assumption of nonnegative coecients (2.4), which we now proceed to do. Fix an 0 and write P(b n R n (; X)6x) 6 Pfor all j = :::; ; 0; ;::: for all xb n 6k6n and all i =0;:::;n k Z j 6k( + ) + P b n R n ()6x; and for some j = :::; ; 0; ;::: Z j k( + ) for some xb n 6k6n and some i =0;:::;n k :=P(A )+P(A 2 ): (2.7) Since P(A )=P M (x;( + ); n) in (2.3), we immediately conclude by Lemma 3.3 that P(A ) = exp{ p( h) ( =h + + =h ) h ( + ) x h }: (2.8) One sees from this that the asymptotic upper bound P(bn R n (; X)6x)6exp{ p( h) ( =h + + =h ) h x h } (2.9) will follow from (2.7) and (2.8) upon letting 0 once we show that for every 0 P(A 2)=0: To this end we dene for a given j and k(j) = arg xb n6k6n k i(j) = arg i=0;:::;n k(j) i=0;:::;n k i+k(j) j : In the case of a tie, choose, say, the smallest possible value of the arg. (2.0)

8 26 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 We consider the rst (from the left) noise variable Z j responsible for the exceedance in the event A 2 by dening T := inf j = :::; ; 0; ;::: Z j k( + ) for some xb n 6k6n and some i =0;:::;n k : It follows from Lemma 3.3 that, at least for large n; T cannot take value ; see Corollary 4:3 in Manseld et al. (999). Therefore, (2.0) will follow once we show that P(A 2 T = j 0 )=0: (2.) j 0 For a given j 0 let k 0 = k(j 0 ) and i 0 = i(j 0 ). As in Manseld et al. (999), we see that P(A 2 T = j 0 ) 6 P + P 0 j Z j j=j 0+ d=i 0+ j j 0 Z j 0 j d=i 0+ j 6 k 0 =2 6 k 0 =2 T = j 0 := P ;j0 (A 2 )+P 2;j0 (A 2 ): (2.2) Recall the well-known inequality (which is an easy consequence of the Burkholder Gundy inequality; see, e.g. Theorem 5:6: in Kwapien and Woyczynski (992)): for any 0 p62, for some absolute constant C p, p n n E Y j 6C p E Y j p (2.3) j= j= for any n and independent zero mean random variables Y ;:::;Y n. Choose now p min(; 2), and x, for now, M. We use Markov s inequality, Jensen s inequality and (2.3) to conclude that P ;j0 (A 2 ) 6 E j=j 0+ Z j ( ) p 2 6 k p 6 Ck p 0 E i0+k 0 j d=i 0+ j d p (k 0 =2) p i 0+k 0 j Z j 0 E Z j 0 j d=i 0+ j d=i 0+ j p ( d 6M) p

9 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) C + Ck p 0 E M d= M + CE Z j p E k 0 i 0+k 0 k 0 d=i 0+ ( 6 C i 0+k 0 E k 0 d=i 0+ 0 j d=i 0+ j 0 d j=i 0+ d ( d M) Z j p Z j j ( d j M) p) =p j ( d j M)Z j p k + CE 0 M Z j k 0 p ; (2.4) j= d= M where the last inequality above is the triangle inequality in L p. Here and in the sequel C is a generic nite positive constant that may be dierent from place to place. By the law of large numbers, the second term in the right-hand side of (2.4) converges to zero, whereas the rst term in the right-hand side of (2.4) is simply ( p) =p C E j ( j M)Z j ; and the latter expression can be made as small as one wishes by letting M. Therefore, P ;j0 (A 2 )=0: (2.5) j 0 We now switch to estimating the second term, P 2;j0 (A 2 ), in the right-hand side of (2.2). Its treatment is similar to that of P ;j0 (A 2 ) above, but is a bit more involved because conditioning forces us to deal with nonzero means. Observe, rst of all, that P 2;j0 (A 2 )=P j 0 Z () j 0 j d=i 0+ j 6 k 0 =2 ; where (Z () j ) are independent random variables such that each Z () j has the law of Z j conditioned on belonging to the interval ( ] k(j) S j = ;( + ) i(j)+k(j) j d=i(j)+ j : d Note that, at least for large n, the conditioning is well dened. We decompose P 2;j0 (A 2 ) 6 P min(0;j 0 ) Z () j 0 j d=i 0+ j p 6 k 0 =6 p p

10 28 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) P + P 6j min(n;j 0 ) n6j6j 0 Z () j Z () j 0 j d=i 0+ j 0 j d=i 0+ j 6 k 0 =6 6 k 0 =6 := P 2;j0 (A 2 )+P 22;j0 (A 2 )+P 23;j0 (A 2 ): (2.6) We estimate P 2;j0 (A 2 ) rst. Let us show that min(0;j 0 ) i k 0 j 0 E 0+k 0 j Z () j d=i 0+ j Indeed, notice that for all n large enough, uniformly in j, EZ () =0: (2.7) j = E(Z Z S j )= E(Z (Z S j )) 2E(Z (Z S j )): (2.8) P(Z S j ) Observe also that EZ () j 0 for all j. Therefore, for all n large enough, we can use Potter s bounds (see, e.g. Proposition 0:8 of Resnick (987)) to see that for any (0; ), E min(0;j 0 ) Z () j min(0;j 0 ) 62 min(0;j 0 ) 6C 0 j d=i 0+ j E(Z (Z S j )) 0 j d=i 0+ j k(j) 0 j d=i 0+ j i(j)+k(j) j d=i(j)+ j : (2.9) By Lemma 3. we can replace the coecients ( ) in (2.9) by a dominating sequence that is nonincreasing (on 0; ; 2;:::). To simplify the notation, we keep calling the new sequence ( ). By the monotonicity for every j60 and i(j)+k(j) j 6 xb n j k(j) xb n 0 j d=i 0+ j d=i(j)+ j d= j 6 k xb n j 0 ; xb n d= j with the usual notation of x and x for, correspondingly, the greatest integer not exceeding x and the smallest integer at least as big as x. Therefore, if, in addition

11 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) of being less than, is also smaller than h, min(0;j 0 ) i k 0 j 0 E 0+k 0 j Z () j 6C d=i 0+ j 0 xb n xb n j d= j as n by the dominated convergence theorem. To see this last statement, note that by monotonicity, for every n, xb n j 6 j ; xb n and 0 d= j j by the choice of. This establishes (2.7). It follows from (2.7) that for all n large enough and p min(; 2), min(0;j 0 ) i 0+k 0 j P 2;j0 (A 2 ) 6 P (Z () j EZ () j ) 6 k 0 =8 d=i 0+ j 6 E min(0;j 0 ) (Z() j EZ () j ) 0 j d=i 0+ j d p (k 0 =8) p : (2.20) It is elementary to check that for some constant C, all n large enough and all j Z () j EZ () j 6 C Z (2.2) in the sense of the usual stochastic domination. Therefore, using once again the inequality (2.3), we conclude that p i P 2;j0 (A 2 )6Ck p 0+k 0 j 0 E Z j ; d=i 0+ j which is one of the intermediate expressions in (2.4). Therefore, we already know that P 2;j0 (A 2 )=0: (2.22) j 0 An essentially identical argument shows that P 23;j0 (A 2 )=0; (2.23) j 0 and, moreover, we see that to prove that P 22;j0 (A 2 ) = 0 (2.24) j 0 as well, we only need to check that i k 0 j 0 E 0+k 0 j Z () j =0: st 6j min(n;j 0 ) d=i 0+ j 0

12 30 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 In order to prove the latter statement, we need, as before, to show that k 0 j 0 n E(Z (Z S j )) j= 0 j d=i 0+ j =0: (2.25) Replacing, once again, the sequence ( ) with a dominating sequence with the same name that is nonincreasing as the subscript moves away from the origin in either direction, we see that for every j i(j)+k(j) j d=i(j)+ j 6 k(j) 6 k(j) d=0 + k(j) k(j) d=0 d k(j) xbn xbn d=0 d=0 d + 6C (xb n )6Ca n xb n xb n for large n, by Karamata s theorem (see, e.g., Theorem 0:6 in Resnick (987)) and (.). Therefore, once again by Karamata s theorem and (.9), for all n large enough and all j E(Z (Z S j ))6C a n n ; and so in order to prove (2.25) we have to show that a n n j 0 Write n n 0 j k 0 j= d=i 0+ j 0 j k 0 j= d=i 0+ j = i 0 j= ( )+ =0: (2.26) i 0+k 0 j=i 0+ ( )+ n j= 0+ ( ):=S (n)+s 2 (n)+s 3 (n) and observe that by the monotonicity of the coecients and Karamata s theorem i 0 n S (n)6 j 6 j 6Cn (n): j= j= Since a n (n) is regularly varying with exponent = h, we immediately conclude by (.6) that Similarly a n n j 0 a n n j 0 S (n)=0: S 3 (n)=0: Finally, S 2 (n)6 0 (k 0 + d) + k 0 (k 0 d) 6 k 0 k 0 d= k 0 d=0 and we obtain, as above, that a n n S 2 (n)=0 j 0 n 6Cn (n); d= n

13 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) as well. Therefore, (2.26) follows, and so (2.24) has been established as well, and it follows from (2.6), (2.22), (2.23) and (2.24) that P 2;j0 (A 2 )=0: (2.27) j 0 However, (2.2), (2.5) and (2.27) establish (2.). That is, (2.0) follows as well, and so we have completed the proof of the asymptotic upper bound (2.9). The proof of the theorem will be complete once we establish the asymptotic lower bound inf P(b n R n (; X)6x) exp{ p( h) ( =h + + =h ) h x h }: (2.28) We have for an (0; ), N max(;x) and 0 P(b n R n (; X)6x) P(B ) P(B2) c P({b n R n () x} B B 2 ); (2.29) where B = for all j = :::; ; 0; ;::: for all xb n 6k6n and all i =0;:::;n k Z j 6k( ) and B 2 = for each xb n6k Nb n and each i =0;:::;n k for at most one j = :::; ; 0; ;::: : Z j b n We have by Lemma 3.3 P(B ) = P M (x;( ); n) = exp{ p( h) ( =h + + =h ) h ( ) x h }: (2.30) Furthermore, by Lemma 3.4, P(Bc 2)=0: (2.3) Therefore, the asymptotic lower bound (2.28) will follow by letting 0ifwecan show that for every (0; ) and 0 small enough comparatively to N P({b n R n () x} B B 2 )=0: (2.32) An application of Lemma 3.5 shows that (2.32) will follow if we prove that for every (0; ), N and 0 small enough comparatively to, P({x b n R n () N} B B 2 )=0: (2.33)

14 32 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 However, the proof of (2.33) is identical to the proof of (3.28) in Manseld et al. (999). Therefore, we have established the matching asymptotic lower bound (2.28), and so we have completed the proof of the theorem in the case of nonnegative coecients and, hence, in the case + ; 0 in (.5). Suppose now that + = 0 and 0, (say). Letting (3) j = j (j 0); (4) j = j (j 0) for j we dene two innite moving averages X (3) and X (4) by X (3) n = (3) n j Z j; X (4) n = (4) n j Z j; n =; 2;:::. Observe that the innite moving average process X (4) has vanishing coef- cients j with j 0 and eventually nonnegative coecients for j 0. Therefore, the decomposition performed at the beginning of the proof shows that in order to extend the claim of the theorem to the process X (4), it is enough to do it under the assumption that j 0 for all j 0, and under this assumption the statement of the theorem has already been established. Furthermore, X n = X n (3) + X n (4) for all n, and an appeal to Lemma 3.2 shows that the statement of the theorem for the original process X will follow once we prove that P(b n R n (; X (3) ) ) 0 (2.34) for every 0. To this end, let 0. Since + = 0 there is a nonnegative sequence ( j ) such that j j for all j 0 and j j (j) = : Dene yet two other innite moving averages X (5) and X (6) by X n (5) = ( n j (3) n j )Z j; X n (6) = n jz j ; n =; 2;:::. Observe that both processes have nonnegative coecients, and that X (3) X n (6) X n (5) for all n. We already know that P(b n R n (=2; X (5) ) )= exp{ q( h) h (=2) h } and P(b n R n (=2; X (6) ) )= exp{ p( h) h (=2) h } (see also Remark 2.3). Therefore, another application of Lemma 3.2 gives us P(bn R n (; X (3) ) ) 6 ( exp{ q( h) h (=2) h }) n = +( exp{ p( h) h (=2) h }): Now let 0 to obtain (2.34). This completes the proof of the theorem in all cases.

15 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) Lemmas and other auxiliary results The rst lemma states several useful properties of regular varying functions. It is well known and is presented here for convenient reference. Lemma 3.. Let g be a nonnegative regularly varying at innity function with a positive exponent of regular variation. (i) The generalized inverse g in (:8) is a nondecreasing regularly varying function with exponent = of regular variation satisfying g(g (y)) g (g(y)) = =: y y y y (ii) There are nondecreasing nonnegative regularly varying at innity functions g and g ; each with exponent of regular variation; such that g 6g6g and g (x) x g (x) =: Proof. Part (i) is Theorem :5:2 in Bingham et al. (987). For part (ii) simply take g (x) = inf g(y); y x g (x) = g(y): 0 y6x See Karamata (962). We continue with an elementary property of the length of long strange intervals dened in (.). Lemma 3.2. Let X =(X ;X 2 ;:::) and Y =(Y ;Y 2 ;:::) be stochastic processes dened on the same probability space. For all real and 2 and n R n ( + 2 ; X + Y)6max(R n ( ; X);R n ( 2 ; Y)): (3.) Proof. The statement (3.) is an elementary consequence of the denition (.). Our next lemma describes the asymptotic behavior of the probability P M (x;; n) in (2.3). We remind the reader that we view P M (x;; n) as the main term in the probability of interest P(b n R n (; X)6x) in Theorem 2.. Note that even though we state the assumptions on the noise variables in the same language as before, i.e. we assume (.3), this is only done for consistency with the rest of the paper. In fact, this lemma does not require any assumptions on the left probability tail of the noise variables. Furthermore, one may allow + = = 0 here, with the convention that the right-hand side of (3.2) is equal to zero in that case. The proof of the lemma is along the same lines as the proof of Lemma 4:2 in Manseld et al. (999). Lemma 3.3. Assume that the coecients ( j ) are nonnegative and satisfy the balanced regular variation assumption (:5). Assume; further; that the noise variables satisfy (:3). Then for all x 0 and 0 P M (x;; n) = exp{ ( h) ( =h + + =h ) h x h }: (3.2)

16 34 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 Proof. Begin by writing P M (x;; n)= P Z6 xb n6k6n xb n6k6n k i=0;:::;n k ; and observe that by Lemma 4: in Manseld et al. (999) the statement (3.2) will follow once we prove that P Z k i=0;:::;n k =( h) ( =h + + =h ) h x h : (3.3) Using Potter s bounds and denition (.9) of a n, we see that (3.3) will, in turn, follow if we show that for all in some neighborhood ( ; + ) of we have n a n xb n6k6n k i=0;:::;n k =( h) ( =h + + =h ) h x h : (3.4) By choosing small enough, and recalling (.6), we see that it is enough to prove (3.4) for =h. We split, as it is usual, the proof of (3.4) into separate proofs of lower and upper bounds. We start with proving the lower bound Let inf n a n xb n6k6n k i=0;:::;n k ( h) ( =h + + =h ) h x h : (3.5) =h = =h + + =h : (3.6) We have, with the usual notation of x and x for, correspondingly, the greatest integer not exceeding x and the smallest integer at least as big as x, n n a n a n xb n6k6n k n ( ) xb n j= xb n i=0;:::;n k xb n i+ xb n j i=0;:::;n xb n

17 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) n a n a n a nb n n ( ) xb n j= xb n xb n x xb n xb n xb n d=0 ( )xb n d=0 xb n xb n d= xb n + xb n xb n xb n + d= d d= d : (3.7) By (.5), (.4) and Karamata s theorem (Theorem 0:6 in Resnick (987)) we have, as n, and ( )xb n d= + ( h) ( )xb n (( )xb n ) xb n d= d ( h) xb n (xb n ); and so the right-hand side of (3.7) is, asymptotically, equal to ( h) a n( + ( ) (( )xb n )+ (xb n )) ( h) x h (a n (b n )) ( + ( ) h + h ) ( h) ( =h + + =h ) h x h by denition (.) of b n and Lemma 3., as required. Hence, lower bound (3.5) has been established, and it remains to prove the corresponding upper bound n a n xb n6k6n k i=0;:::;n k 6( h) ( =h + + =h ) h x h : (3.8) By the obvious monotonicity argument it is enough to prove (3.8) in the case + 0; 0 and, in the latter case, by the second part of Lemma 3. we may assume that the function ( ) and both sequences of coecients ( j ;j 0) and ( j ;j 0) are nonincreasing. Split the sum over j in the left-hand side of (3.8) into three sums: over j60, over 6j6n and over j n, and denote the corresponding sums by S (n), S 2 (n) and S 3 (n). We will prove that n S i(n)=0; for i =; 3; (3.9)

18 36 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 and n S 2(n)6( h) ( =h + + =h ) h x h : (3.0) By the monotonicity of the coecients ( j ;j 0) we have k j xb 0 0 S (n) = a n n j = a n k xb n 6 x a nb n 6 x a nb n xb n6k6n xb n j=0 d= j xb n +j d=+j xb n 2 xb n d= + + j= xb n + j= xb n + xb n +j d=+j (xb n ) j+ 6 Ca nb n (b n (b n (2 xb n )) + b n(b n ( xb n +) )) 6 Cb n (a n (b n )) Cb n d= j as n, where the last asymptotic equivalence follows from denition (.) of b n. Here C is a nite positive constant (in the sense of independence of n; C may depend, for instance, on x, h and ). Recall that C does not have to be the same every time it appears (the same convention about unspecied nite positive constants holds throughout the paper). The computation above used regular variation of, Karamata s theorem (Theorem 0:6 in Resnick (987)) and the fact that h. Since (b n ) is regularly varying with exponent =(h), we obtain (3.9) with i =. Furthermore, (3.9) with i = 3 follows from (3.9) with i = upon changing names by j = n + j, i = n k i and d = d. Hence, we only need to prove (3.0) to complete the proof of the upper bound (3.8) and, hence, the proof of the lemma. To this end, let 0 h, and choose M = M() so large that D 6( + )( h) + D (D); d=0 D d 6( + )( h) D (D) d=0 (3.) for all D M and (kr)6( + )r h (k) (3.2) for all M=k6r and all k large enough. Existence of such an M is guaranteed by Karamata s theorem and Potter s bounds. Write for j = ;:::;n m j := k = max{m j; ;m j;2 ;m j;3 }; xb n6k6n i=0;:::;n k

19 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) where m j; = m j;2 = xb n6k6n xb n6k6n k k i=0;:::;n k i j M i=0;:::;n k i j k+m ; and m j;3 = xb n6k6n k i=0;:::;n k j k+m6i6j M : Observe that by the monotonicity of the coecients m j; 6 xb n + (3.3) xb n and m j;2 6 xb n d= (M ) M d=0 xb n + d= d=0 d : (3.4) Furthermore, for every k xb n and i =0;:::;n k; j k + M6i6j M, let r = j i [ M k k ; M ] k and note that by (3.) and (3.2) for all n large enough, = d=0 j i + d= d 6 ( + )( h) ( + (i + k j) (i + k j) + (j i ) (j i )) 6 ( + ) 2 ( h) k (k)( + ( r) h + r h ): The function of r [0; ] above achieves its maximum at r = =(h+) =(h+) + + =(h+) and, upon substitution, one has 6( + ) 2 ( h) ( =(h+) + + =(h+) ) h+ k (k):

20 38 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 Therefore, for all n large enough, and all j =;:::;n m j;3 6( + ) 2 ( h) ( =(h+) + + =(h+) ) h+ ( xb n ); (3.5) and the maximization procedure performed above, together with (3.3) and (3.4) also shows that for all n large enough, and all j =;:::;n m j;l 6( + ) 2 ( h) ( =(h+) + + =(h+) ) h+ ( xb n )+ C xb n ; l=; 2 (3.6) for a nite constant C = C(M). Therefore, for every 0 h n S 2(n) = n n (a n m j ) j= 6 ( + ) 2 ( h) ( =(h+) + + =(h+) ) (h+) (a n ( xb n )) =(+) 2 ( h) ( =(h+) + + =(h+) ) (h+) x h : Now let 0 to obtain (3.0). This completes the proof of the lemma. Our next lemma is a result parallel to Lemma 4:4 in Manseld et al. (999). The dierent circumstances necessitate a dierent argument. Even though we assume the usual assumptions in force throughout the paper, the assumptions on the tail of the noise variables and on the balanced regular variation of the coecients may be relaxed through a simple domination argument. Lemma 3.4. Assume that the coecients ( j ) are nonnegative and satisfy the balanced regular variation assumption (:5). Assume; further; that the noise variables satisfy (:3). Then for all M 0 and 0 for some k =;:::; Mb n and some i =0;:::;n k P Z j b n for at least two dierent j = :::; ; 0; ;::: =0: (3.7) Proof. The nonnegativity of the coecients shows that (3.7) will follow once we prove that for all M 0 and 0 n j for some i =0;:::;n Z j b n P for at least two dierent j = :::; ; 0; ;::: =0: (3.8)

21 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) Notice that by Lemma 3. we may (and, as usual, will) assume that the coecients ( ) are nonincreasing for d 0 and nondecreasing for d60. By the monotonicity of the coecients, we see that for every K n j P for some i =0;:::;n Z j b n for some j60 6 ( P Z j j=0 = o() + b n j+ Mbn d=j+ ( P Z j j=k Since the sum P(Z j c j ) j= ) b n j+ Mbn d=j+ ) 6o() + j=k ( P Z j ) M j : converges for all c 0, we immediately conclude that n j for some i =0;:::;n Z j b n for some j60 =0: P An identical argument shows that for some i =0;:::;n P for some j n + Mb n =0: n j Z j b n Therefore, (3.8) will follow once we show that for some i =0;:::;n P n j Z j b n for at least two dierent j =;:::;n+ Mb n =0: (3.9) To this end, note that the probability in (3.9) is bounded from above by n j P for some i =0;:::;n Z j b n for at least two dierent j =;:::;i

22 40 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) P for some i =0;:::;n n j Z j b n for at least two dierent j = i +;:::;i+ Mb n + P for some i =0;:::;n n j Z j b n for at least two dierent j = i + Mb n +;:::;n+ Mb n + P for some i =0;:::;n and n m d=i+ m + P for some i =0;:::;n and n m d=i+ m n j Z j b n for some j =;:::;i Z m b n for some m = i +;:::;i+ Mb n n j Z m b n for some m = i + Mb n +;:::;n+ Mb n + P for some i =0;:::;n n j for some j = i +;:::;i+ Mb n and Z j b n for some j =;:::;i Z j b n n m d=i+ m for some m = i + Mb n +;:::;n+ Mb n Z m b n 6 := p l (n): l= (3.20)

23 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) Observe that by the monotonicity of the coecients Mb n p (n)6p for some j =;:::;n Z j b n ; d= and for some m =;:::;j Mb n +j m d=+j m Z m b n : Let { } T = max j6n: Z j b n Mbn d= : Then ( ) n b n p (n)6 P(T = j)p Z m Mbn +j m j=2 d=+j m for some m =;:::;j : d Now for every K, for every j =2;:::;n ( ) P b n Z m Mbn +j m d=+j m for some m =;:::;j ( ) 6KP Z b n Mbn d= ( ) + P b n Z i Mbn +i for some i = K +;:::;j d=+i ( ) 6KP Z b ( n Mbn + P Z i ) d= M i for some i = K +;:::;j ( ) 6KP Z b ( n Mbn + P i Z i ) ; d= M i=k+ and so p (n)6 i=k+ ( P i Z i ) : M Since the sum in the right-hand side above can be made as small as we wish by choosing K large enough, we conclude that p (n)=0: An identical argument shows that p 3(n)=0: (3.2) (3.22)

24 42 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 Furthermore, by the nonnegativity of the coecients p 2 (n) 6 P for some j =;:::;n+ Mb n Let Mb n d= ( Mb n ) and for some m = j +;:::;j+ Mb n Mb n Z m b n : d= ( Mb n ) T 2 = min j : Z b n j Mbn d= ( Mb n ) : Z j b n Then by Karamata s theorem (Theorem 0:6 in Resnick (987)) and (.), n+ Mb n p 2 (n) 6 P(T 2 = j) j= b n P Z m Mbn d= ( Mb for some m = j +;:::;j+ Mb n n ) d b n 6 Mb n P Z Mbn d= ( Mb 6Mb n P(Z C (b n ) ) n ) d 6 Mb n P(Z Ca n )6 b n n 0 as n by (.6). Therefore, p 2(n)=0: For every K we have p 4 (n) 6 P (for some i =0;:::;n n j Z j b n for some j = i Kb n ;:::;i and n m d=i+ m Z m b n (3.23) for some m = i +;:::;i+ Mb n ) + P for some j =;:::;n+ Mb n n j Z j b n

25 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) for some i j+ Kb n := p 4 (n)+p 42 (n): (3.24) An argument identical to that used to establish (3.23) shows that for every K p 4(n)=0: (3.25) On the other hand, by the monotonicity of the coecients and (.), b n p 42 (n) 6 (n + Mb n )P Z Kbn + Mb n d= Kb n + 6 (n + Mb n )P (Z M ) ( Kb n ) and so 6 (n + Mb n )P(Z CK h (b n ) ) 6 (n + Mb n )P(Z CK h a n )6C n + Mb n K h ; n p 42 (n)6ck h ; which can be made arbitrarily small by choosing K large enough. Therefore, p 42(n)=0; (3.26) and we conclude by (3.24), (3.25) and (3.26) that p 4(n)=0: (3.27) An identical argument shows also that p 6(n)=0: (3.28) Therefore, (3.9) (and, hence, the statement of the lemma) will follow once we treat the last remaining term in the decomposition (3.20) and show that p 5(n)=0: (3.29) This is easy, given what we already know. For a K write n j p 5 (n) 6 P for some i =0;:::;n Z j b n for some j = i Kb n ;:::;i and n m d=i+ m Z m b n for some m = i + Mb n +;:::;i+ Mb n + Kb n

26 44 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) P for some j =;:::;n+ Mb n for some i j+ Kb n n j Z j b n + P for some j =;:::;n+ Mb n n j Z j b n for some i j Mb n Kb n :=p 5 (n)+p 52 (n)+p 53 (n): The same argument as that used to prove (3.23) shows that for every K p 4(n)=0; whereas p 52 (n)=p 42 (n), and so p 52 (n)6ck h : An identical argument shows that p 53 (n)6ck h as well. Letting K we conclude that p 52(n) = p 53(n)=0; and so (3.29) follows. This completes the proof of the lemma. The last lemma is the analog of Lemma 4:8 in Manseld et al. (999), and it can be proven in the same way as the latter. Indeed, the key ingredient in the proof of Lemma 4:8 in Manseld et al. (999) is Lemma 4:6 in the latter paper, and the assumptions of that lemma are satised under the setup of the present paper. Lemma 3.5. Assume that the coecients ( j ) are nonnegative and satisfy the balanced regular variation assumption (:5). Assume; further; that the noise variables satisfy (:3). Then M P(R n (; X) Mb n )=0: 4. Numerical examples and issues There are many practical issues that have to be resolved before the iting theorems proved in this paper and in Manseld et al. (999) can be used as the basis of a statistical procedure to detect and test for long-range dependence. The most important

27 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) Fig observations from the moving average process with = 3 and h =:35 (the left plot) and with =:5 and h =:3667 (the right plot). issues are the rate of convergence in the iting theorems and the choice of the parameter. Even though the iting distributions in Theorem 2. of the present paper and Theorem 2. of Manseld et al. (999) do not involve, it is clear the value of the statistics for nite samples will be aected by the choice of and, perhaps, signicantly so. These issues are the subject of ongoing research. Here we present a modest empirical study that is designed to illustrate the above issues. We simulated observations from a moving average process of the form X n = n j h Z j ; n=; 2;::: ; (4.) j n rst with (Z j ) being iid symmetric -stable random variables with =:5 and scale = (see, e.g. Samorodnitsky and Taqqu (994)), and then with (Z j ) being iid symmetrized Pareto random variables with =3. In the latter case, P(Z j x)=p(z j x)=x 3 =2 for x. We truncated the innite series in (4.) to the terms with j n 6250; 000. The simulations with =:5 were done for h ranging from h =0:7667 to h =:9667 with step of 0:. For = 3 we used the range of h from h =0:55 to h =2:05 with step of 0:. To give the reader the avor of the observations, the plots in Fig. display the rst 000 observations for = 3 and h =:35 and for =:5 and h =:3667. Next, for =:5 we calculated the lengths of the longest strange segments for =; :5; 2 and 3:5. For =3 we calculated the lengths of the longest strange segments for =3:5; 4:25; 4:75 and 5:25. Once again, to give the reader the avor of the behavior of the lengths of the longest strange segments as a function of the sample size, the plot in Fig. 2 below presents this function as measured in our simulations for =3; h=:35 and =4:75 and for =:5; h=:3667 and =3:5. For each choice of and h we estimated the rate of growth of the logarithm of the length of the longest strange segment as a function of the logarithm of the sample size by computing the ratio of the two for the entire sample (n = ). We plot the results for each as a function of h, together with the theoretical slope =(h) if h and = if h. The rst plot, that in Fig. 3 is for =3. Table presents the same data in the tabular form. Note that for the sample size we are using, our estimates are reasonably accurate for h, but biased upwards for h.

28 46 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 Fig. 2. The length of the longest strange segment as a function of the sample size (in units of 0 3 ) for =3, h =:35 and =4:75 (the left plot) and for =:5, h =:3667 and =3:5 (the right plot). Fig. 3. Estimated slope of log(r n(; X)) as a function of log n for = 3. Horizontal axis is h. Number 2 corresponds to =3:5, number 3 corresponds to =4:25, number 4 corresponds to =4:75 and number 5 corresponds to = 5:25. Number is the slope predicted by the it theorems. Table Estimated slope of log(r n(; X)) as a function of log n for =3 Slope estimate h Theoretical =3:5 =4:25 =4:75 =5:

29 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) Fig. 4. Estimated slope of log(r n(; X)) as a function of log n for =:5. Horizontal axis is h. Number 2 corresponds to =, number 3 corresponds to =:5, number 4 corresponds to = 2 and number 5 corresponds to = 3:5. Number is the slope predicted by the it theorems. Table 2 Estimated slope of log(r n(; X)) as a function of log n for =:5 Slope estimate h Theoretical = =:5 =2 =3: The plot in Fig. 4 gives the rate of growth of the logarithm of the length of the longest strange segment as a function of the logarithm of the sample size for =:5. Finally, Table 2 presents the same data in the tabular form. It is clear from the data we presented here that the problem of selecting the appropriate level and the connection between such level and the sample size is a dicult one, that has to be seriously addressed. Acknowledgements The plots in this paper are due to Borjana Racheva, whose help is gratefully acknowledged. The authors are grateful to Carlo Marinelli and Boris Hristov for their help in the process of data simulation, analysis and handling.

30 48 S.T. Rachev, G. Samorodnitsky / Stochastic Processes and their Applications 93 (200) 9 48 References Beran, J., 994. Statistics for Long-Memory Processes. Chapman and Hall, New York. Bingham, N., Goldie, C., Teugels, J., 987. Regular Variation. Cambridge University Press, Cambridge. Brockwell, P., Davis, R., 99. Time Series: Theory and Methods, 2nd Edition. Springer, Berlin, New York. Dembo, A., Zeitouni, O., 993. Large Deviations Techniques and Applications. Jones and Bartlett, Boston. Hurst, H., 95. Long-term storage capacity of reservoirs. Trans. Amer. Soc. of Civ. Eng. 6, Karamata, J., 962. Some theorems concerning slowly varying functions. Mathematics Research Center Technical Report No. 369, University of Wisconsin, Madison. Kwapien, S., Woyczynski, N., 992. Random Series and Stochastic Integrals: Single and Multiple. Birkhauser, Boston. Manseld, P., Rachev, S., Samorodnitsky, G., 999. Long strange segments of a stochastic process and long range dependence. Preprint. Avalilable as rareseg.ps.gz at Mikosch, T., Samorodnitsky, G., The remum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 0, Resnick, S., 987. Extreme Values, Regular Variation and Point Processes. Springer, Berlin, New York. Samorodnitsky, G., Taqqu, M., 994. Stable Non-Gaussian Random Processes. Chapman and Hall, New York.