THE HURST INDEX OF LONG-RANGE DEPENDENT RENEWAL PROCESSES. By D. J. Daley Australian National University

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1 The Annals of Probabiliy 1999, Vol. 7, No. 4, THE HURST INDEX OF LONG-RANGE DEPENDENT RENEWAL PROCESSES By D. J. Daley Ausralian Naional Universiy A saionary renewal process N for which he lifeime disribuion has is kh momen finie or infinie according as k is less han or greaer han κ for some 1 <κ<, is long-range dependen and has Hurs index α = 1 3 κ (his is he criical index α for which lim sup a var N is finie or infinie according as a is greaer han or less han α). This idenificaion is accomplished by delineaing he growh rae properies of he difference beween he renewal funcion and is linear asympoe, hereby exending work of Täcklind. 1. Inroducion and saemen of resuls. The quesion simulaing his noe concerns he idenificaion of he Hurs index of a long-range dependen (LRD) saionary renewal process (see below for definiions). Answering his quesion led us o revisi work of Täcklind on he order of he difference beween he renewal funcion and is linear asympoe when a generic lifeime of he renewal process has infinie second momen, which is a necessary and sufficien condiion for he saionary renewal process o be LRD. Hisorically [we refer he reader o Beran (1994) for a broad accoun] longrange dependen processes X n in discree ime evolved in work of he hydrologis Hurs (1951) who sudied annual flow daa of he River Nile. This moivaed Mandelbro and coworkers o inroduce fracional Gaussian noise as a saisical model wih long memory. Landmarks in he saisical lieraure include Taqqu (1986) and, especially, Beran (1994). Adapaion of he idea o he poin process seing, for second-order saionary poin processes, is in Daley and Vesilo (1997). A saionary renewal process wih couning funcion N A denoing he number of poins in he Borel se A is characerized by he disribuion funcion (d.f.) F of a generic lifeime random variable (r.v.) X which has finie mean λ 1 = E X = xdf x. For convenience we assume ha = F + = Pr X = and wrie N when A is he half-open inerval. N is longrange dependen (LRD) in he sense ha (1.1) lim sup var N = if and only if E X =, because his is he necessary and sufficien condiion for he inegrand in he represenaion [e.g., Daley and Vere-Jones (1988), Received November 1998; revised July AMS 1991 subjec classificaions. Primary 6K5; secondary 6G55. Key words and phrases. Hurs index, long-range dependence, renewal process, renewal funcion asympoics, regular variaion, momen index. 35

2 36 D. J. DALEY Chaper 3, Secion 5 and equaion (4.3.4)] ( ) (1.) V var N =λ U u λu 1 du o diverge o as [e.g., Täcklind (1944), Feller (1966), Chaper XI, Secion 4, or (1971), Chaper XI, Secion 3]; in (1.), U n= F n, where F n is he n-fold convoluion of F wih iself, is he renewal funcion; recall he elemenary renewal heorem U / λ. A saionary poin process N wih finie second momen necessarily has E ( N ) < C (all 1) for some finie C, soishurs index α is well defined by { } var N (1.3) α inf a: lim sup < a and lies in he inerval α 1. For a LRD renewal process, i follows from (1.) and he subaddiiviy of he renewal funcion U + u U +U u ( u > ) [e.g., Daley (1978)] ha U λ and hence ha V λ also, and so is Hurs index lies in he inerval 1 1. Indeed, in erms of is order for large, he variance funcion V has he same asympoic behavior as he funcion (1.4) Ṽ V +λ = λ U u λu du The characerizaion we give of he Hurs index of a saionary LRD renewal process is in erms of he momen index { } (1.5) κ sup k: µ k E X k = x k df x = x k 1 F x dx < where F v =1 F v v [in (1.5), k is any real greaer han or equal o ]. Theorem 1. A saionary renewal process wih lifeime disribuion funcion F which has x df x =, xdf x < and momen index κ, is long-range dependen and has Hurs index α = 1 3 κ. The new par of his saemen is he idenificaion of he Hurs index. This idenificaion has cerainly been known in cases such as hose where he lifeime d.f. F has a regularly varying ail, when raher more is hen known as in Example 1. Example 1 (Renewal process whose lifeime d.f. has a regularly varying ail.) From, for example, Feller [(1971), Chaper XI, Secion 3 or Bingham, Goldie and Teugels (1987), Secion 8.6.1], he lifeime d.f. F has a regularly varying ail, meaning (1.6) 1 F x x κ l x x

3 HURST INDEX OF LRO RENEWAL PROCESSES 37 where 1 <κ< and l is any slowly varying funcion, if and only if he discrepancy funcion φ U λ saisfies (1.7) φ λ κ l / κ κ 1 Since regular variaion is preserved under inegraion [Bingham, Goldie and Teugels (1987), Proposiion 1.5.8], i follows from (1.4) ha when (1.6) holds, (1.8) Ṽ =λ φ u du λ 3 3 κ l 3 κ κ κ 1 From hese relaions we can read off a range of possible behaviors for Ṽ, wih α = 1 3 κ in all cases, consisen wih Theorem 1: 1. If µ κ < hen x 1 l x dx < and hence l x (x ) because l is slowly varying [Proposiion 1.5.9bof Bingham, Goldie and Teugels (1987)]. Then Ṽ =o 3 κ in his case.. We can have boh µ κ = and l x, so ha Ṽ =o 3 κ is possible when µ κ =. 3. We can also have boh µ κ = and l x for large x, in which case Ṽ >O 3 κ, hough Ṽ =o 3 κ+ε for arbirarily small posiive ε. Our mehod of proof enables us o recover Täcklind s resul ha φ = U λ = o κ when µ κ < and κ< and o elucidae furher properies of he funcion φ as in Theorem. Like Täcklind, our mehods are wholly real variable, and do no involve any ransform calculus. Example in Secion 3 shows ha we canno weaken he resul. Theorem. For a renewal process wih finie mean lifeime λ 1, infinie second momen and momen index κ, as and for arbirarily small posiive ε, (1.9) { = o U λ κ µ κ < and κ< <o κ+ε µ κ = and in eiher case, (1.1) lim sup U λ κ ε = Noe ha in he laer case of (1.9), i follows from a resul of Vuilleumier a Theorem.3.6 of Bingham,Goldie and Teugels (1987) ha here exiss a slowly varying funcion l such ha (1.11) U λ / κ l

4 38 D. J. DALEY. Proofs. I is a sandard resul [e.g., Feller (1966), Chaper XI, Secion 4] ha he discrepancy funcion φ U λ saisfies he equaion (.1) φ =λ + φ u df u and hence, from he soluion of such a general renewal equaion [e.g., Feller (1966) Chaper XI, equaion (1.4)], (.) φ = = du u u U U v + U U ( min v ) by subaddiiviy of U Now he elemenary renewal heorem implies ha U <A+ B (all ) for some posiive A and B, so φ A κ κ (.3) [ ( κ 1 ( v ) ] κ + B min v v) κ 1 When >µ κ = vκ 1, we can appeal o he dominaed convergence heorem o conclude ha he second inegral in 3 as, while he erm involving he oher inegral approaches by inspecion. This proves Täcklind s resul, he firs par of (1.9). Irrespecive of he finieness of µ κ,wehaveµ κ ε < for posiive ε<κ and so for arbirarily small posiive ε, we have by a similar argumen ha for <ε<κ 1, as, φ A κ+ε κ+ε (.4) [ ( κ 1 ε ( v ) ] κ+ε + B min v v) κ 1 ε Boh pars of (1.9) are proved, and by subsiuion in (1.4) i follows ha α 1 3 κ. To prove (1.1) and ha α 1 3 κ, we argue by conradicion. Suppose ha (1.1) does no hold, ha is, ha for some posiive ε and C as above (1.3), φ C κ ε for all sufficienly large. Then for finie posiive T, φ (.5) T d 1 ε/ d T 3 κ ε/ T From he equaliy in (.) and he inequaliy U λ, his inegral cerainly dominaes (.6) T λ κ ε/ d = T v κ 1+ε/ T κ 1+ε/ F v κ 1 + ε/ dv =

5 HURST INDEX OF LRO RENEWAL PROCESSES 39 by definiion of κ. Hence a conradicion, so (1.1) mus hold, and Theorem is proved. Jus as φ saisfies a general renewal equaion, so, oo, does Ṽ defined a (1.4), namely, s Ṽ =λ ds + λ ds φ s u df u s (.7) [ ] = λ v + + Ṽ u df u so, appealing again o he soluion of he general renewal equaion as for (.1), (.8) Ṽ =λ du u = λ ds s u ds du u s s = λ U s ds = λ U s ds = λ U s ds s v + [ ] λ 3 v v + since U s λs s Suppose now ha α< 1 3 κ, so ha for some posiive ε, Ṽ <C3 κ ε for all sufficienly large and some finie posiive C. Then, much as a (.5), for sufficienly large posiive T, Ṽ (.9) T d C 1 ε/ d T 4 κ ε/ T However, he lef-hand side here cerainly dominaes λ 3 = T κ ε/ (.1) λ 3 v κ 1+ε/ T κ 1+ε/ F v dv = κ 1 + ε/ T he conradicion again coming from he definiion of κ. So we canno have α< 1 3 κ, and Theorem 1 is proved. 3. Exampleand discussion. Example 1. This example shows ha wihin he class of d.f. s wih specified momen index κ, we canno refine he conclusions of Theorems 1 and, such as occurs in he conex of Example 1, unless we impose some furher condiion on he d.f. F. Specifically, we describe a disribuion for which (cf. Theorem ) lim inf φ / κ =, lim sup φ / κ =. A similar example can be consruced for Theorem 1. The genesis of he paricular example is in he Example o Case I of Theorem.6.6 in Bingham, Goldie and Teugels (1987).

6 4 D. J. DALEY Consider a lifeime disribuion F ha consiss of aoms of size Cp n /xn 3/ locaed a he poins x n, n = 1 where hese poins and he weighs p n are so chosen ha p n (n bu n=1 p n /x ε n < for every posiive ε, no maer how small, and C is a normalizing consan. Then he disribuion Cp n /x 3/ n has momen index κ = 3. In he seing of Theorem we recall (.), so we are ineresed in (3.1) φ 1 v= 1/ 1/ ( v = 1 F v d ( 1 v) + 1/ ) 1/v 3/ df v + 1 ( 1/ ( v) ) v 3/ df v v For he discree disribuion we are considering, hese wo inegrals yield finie and infinie sums, respecively. For given, we can find n such ha x n < x n+1. Suppose he p n are such ha he wo sums a 3.1 are dominaed by he erms in p n and p n+1, respecively; a sufficien condiion for his is ha x n+1 / x n p n sufficienly fas. Then for such we have φ C 1/ ( xn ) 1/pn ( ) 1/( + ) p x n+1 x n+1 n+1 This expression approaches for close o eiher x n or x n+1, and he expression approaches for x n x n+1. When p n = n and x n = exp n he sequences p n and x n saisfy he condiions we require. This counerexample and, indeed, Theorem 1 (which can be phrased in he form of Theorem ), sugges why he verificaion of long-range dependence and esimaion of α can be fraugh wih difficulies. Suppose λ = 1 and κ = 1 5, hence α = 75. Then, if insead of esimaing (and ploing) log ( Ṽ / 1 5), we use he index 1.5, he funcion is muliplied by = exp log 1 + log when he exponen is small, such as = e ; ha is, he error need no be deecable using a million observaions (and even wih a billion = 1 9 e 1 he same conclusion may hold). Acknowledgmens. This work arose ou of discussion wih Rein Vesilo concerning a sequel o Daley and Vesilo (1997). I hank him and Nick Bingham for discussion. REFERENCES Beran, J. (1994). Saisics for Long-Memory Processes. Chapman and Hall, New York. Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variaion. Cambridge Univ. Press. Daley, D. J. (1978). Upper bounds for he renewal funcion via Fourier heoreic mehods. Ann. Probab Daley,D.J.andVere-Jones, D. (1988). An Inroducion o he Theory of Poin Processes. Springer, New York. Daley, D. J. and Vesilo, R. (1997). Long range dependence of poin processes, wih queueing examples. Sochasic Process Appl Feller, W. (1966). An Inroducion o Probabiliy Theory and Is Applicaions. Wiley, New York.

7 HURST INDEX OF LRO RENEWAL PROCESSES 41 Feller, W. (1971). An Inroducion o Probabiliy Theory and Is Applicaions, nd ed. Wiley, New York. Hurs, H. E. (1951). Long-erm sorage capaciy of reservoirs. Trans. Amer. Soc. Civil Engineers Täcklind, S. (1944). Elemenare Behandlung vom Erneuerungsproblem für den saionären Fall. Skand. Akuarieidskr Taqqu, M. S. (1986). A bibliographical guide o self-similar processes and long-range dependence. In Dependence in Probabiliy and Saisics (E. Eberlein and M. S. Taqqu, eds.) Birkhauser, Boson. School of Mahemaical Sciences Ausralian Naional Universiy Canberra A.C.T. Ausralia daryl@mahs.anu.edu.au