STRONG LAWS OF R/S STATISTICS WITH A LONG-RANGE MEMORY SAMPLE

Transcription

1 Statistica Siica 5(2005), STRONG LAWS OF R/S STATISTICS WITH A LONG-RANGE MEMORY SAMPLE Zhegya Li Zhejiag Uiversity Abstract: I this paper, we show a law of the iterated logarithm for rage statistics with a log-rage memory sample. With the help of this result, usig R/S statistics, we give a test for log-rage depedece. Key words ad phrases: Law of the iterated logarithm, log-rage memory, R/S statistics.. Itroductio Let X, be a sequece of radom variables. Take X = / X j, the adjusted rage of partial sums S 2 () = (X j X ) 2, (.) k R() = max (X j X ) k ad the self-ormalized rage mi k k (X j X ), (.2) Q() = R()/S(). (.3) The Q is called the rescaled rage or R/S statistic. Itroduced by Hurst (95) whe he studied hydrology data of the River Nile, it plays a importat role i testig statistical depedece of a sequece. With referece to it, Madelbrot itroduced a class of Gaussia processes fractioal Browia motios (cf., Madelbort ad Va Ness (968)). The statistic has bee used, for example, i modelig stock prices (cf., e.g., Lo (99) ad Williger, Taqqu ad Teverovsky (999)). Moreover, it is a importat tool i studyig fractal theory ad chaos pheomea. Feller (95) gave the limit distributio of R()/ for a i.i.d. sequece with EX 2 <. Mora (964) cosidered the case of heavy tails. Li (200)

2 820 ZHENGYAN LIN studied laws of the iterated logarithm (LIL) for R/S statistics uder i.i.d. ad mixig samples. Li ad Lee (2002) exteded these results to the AR() model with a i.i.d. sample. But, for this statistic, oe has more iterest i a sample with log-rage memory. May practical models, such as ecoomic time series for stock prices, exhibit log-rage depedece. Lo (99) cause a sequece lograge depedet of order α if the process has a autocovariace fuctio γ k such that k 2α 2 L(k) for α ( γ(k) 2, ) k 2α 2 L(k) for α (0, 2 ) as k, (.4) where L(k) is a slowly varyig fuctio at ifiity ( a k b k meas that a k /b k as k ). A example is the fractioal-differece process X, defied by ( L) d X = ε, (.5) where L is the lag operator, d ( /2, /2) ad ε is white oise. The autocovariace fuctio here is γ k ck 2d as k. The sequece is statioary ad ivertible ad exhibits a uique kid of depedece that is positive or egative, depedig o whether d is positive or egative (Hoskig (98)). Some methods for detectig log-rage memory have bee developed. To this ed, Lo (99) showed a fuctioal cetral limit theorem for R/S statistics (i fact, he cosidered modified R/S statistics) geerated by a zero-mea statioary Gaussia sequece uder a log memory assumptio usig a weak ivariace priciple due to Taqqu (975). A test for α follows: if α (/2, ), the R/S statistics diverge i probability to ifiity; if α (0, /2), the statistics coverge i probability to zero. I either case, the probability of rejectig the ull hypothesis of short-rage memory approaches uity for all statioary Gaussia processes satisfyig (.4). There are may refereces o log memory, such as papers by Grager ad Joyeux (980), Cox (984) ad Bera (992), ad moographs by Bhattacharya ad Waymire (990) ad Bera (994). I this paper, we first show a LIL for rage statistics R() give a lograge memory sample. The, with the help of this result, we develop a test for log-rage depedece. 2. Theorems ad Their Proofs Let S = X j, σ 2 = Var S ad α = 2 2α 2 2. Theorem 2.. Let X, be a statioary sequece of Gaussia radom variables with mea zero ad autocovariace fuctio (.4). The lim sup 2σ 2 log log R α a.s.. (2.)

3 STRONG LAWS OF R/S STATISTICS 82 If, i additio, α L() /2 ( m α L(m) /2 lim sup L( m) ) 0, as /m, (2.2) L() 2σ 2 log log R = α a.s.. (2.3) Remark 2.. We eed a property of slowly varyig fuctios. Let l(x) be a slowly varyig fuctio at ifiity with b = b(x) 0, as x. The for ay δ > 0, l(bx) lim bδ x l(x) = lim l(x) bδ x l(bx) = 0 (2.4) (cf., Hoskig (98)). Usig this property, the fact that for some β > α, L( m) L() ( ) m β (2.5) implies (2.2). Moreover, if there is a β > 0 such that x β L(x) is o-icreasig, the (2.5) is satisfied with β =. Clearly, (2.5) is quite a weak coditio, the commo slowly varyig fuctios, such as log x ad / log x, satisfy it. Usig Theorem 2., we give a test for log-rage memory. Theorem 2.2. Uder (2.2), as, a.s. for α ( 2 Q, ), log log 0 for α (0, 2 ). Remark 2.2. This theorem establishes a test for log-rage memory i the a.s. covergece sese, followig Lo (99) i the weak covergece sese. Remark 2.3. For some cases of short-rage memory, we have established laws of the iterated logarithm for R/S statistics. For istace, for a i.i.d. sequece (correspodig to the case of α = /2) with EX = 0 ad EX 2 =, we have lim sup 2 log log Q = a.s.. For the AR() model with a i.i.d. sample, we have a similar result. I order to prove Theorem 2., we eed some prelimiaries. The followig Ferique-type iequality is a cosequece of Lemma 2. i Csáki, Csörgő ad Shao (992).

4 822 ZHENGYAN LIN Lemma 2.. Let B be a separable Baach space with orm ad let Γ(t), < t < be a stochastic process with values i B. Let P be the probability measure geerated by Γ( ). Assume that Γ( ) is P-almost surely separable with respect to, ad that for t t 0 ad 0 < x x there exists a o-egative odecreasig fuctio σ(x) such that P Γ(t) xσ(t) K exp γx β for some K, γ, β > 0. The P sup 0 t T Γ(t) xσ(t) + σ (T, k) 4K2 2k+ exp γx β for ay 0 < t t 0, x x ad k 3, where σ (T, k) = 4(4/γ) /β β 2 (k 2)/β σ(t e yβ )dy. Usig a discrete versio of this lemma, we ca show the followig strog law of large umbers. Lemma 2.2. For X defied i Theorem 2., we have (/) X j 0 a.s. Proof. By (.4) ad Lemma 3. i Taqqu (975), we have that for α (/2, ), σ 2 = γ(i j) a 2α L(), i= where a = (2α 2α ). Cosider the case of α (0, /2). Put R = γ(0) + 2 k= γ(k). The σ 2 = γ(0) + (R 2 k= j+ γ(k)). Hece R 0. If R = 0, similarly to the proof of Lemma 2. i Taqqa (975), we have also σ 2 a2α L(). (2.6) If R > 0, it is clear that σ 2 R. Cosider the case of σ 2 a 2α L(). By the property of a slowly varyig fuctio, for ay ε > 0, oe was lim x x ε L(x) =, lim x x ε L(x) = 0, 2 /σ 2 c2( α) /L() c α for large, where c stads for a positive costat. Hece, usig Lemma 2. with σ() = σ, for ay ε > 0 there exists a costat C = C(ε) > 0 such that P max S j 2 k ε j 2 k C exp 22k σ 2 2 k ε2 C exp 3 c2k( α) ε 2 which, i combiatio with the Borel-Catelli lemma, implies that as k, max j 2 k S j /2 k 0 a.s. Usig this result, alog the lies of a proof of a strog law of large umbers, we obtai Lemma 2.2. The details are omitted. As for the case of σ 2 R, Lemma 2.2 is the Strog Law of Large Numbers uder oly momet restrictios (cf., Serflig (970)). 3,

5 STRONG LAWS OF R/S STATISTICS 823 Lemma 2.3. For ay ε > 0, there exists C = C (ε) > 0 such that for ay x ad large, P R() /σ > ( + ε) α x C exp ( + ε)x 2 /2. Proof. Rewrite R() as R() = max i<j S j S i ((j i)/)s. Cosider the case σ 2 a 2α L(). For i < j, we have E(S j S i ) 2 = ESj i 2 a(j i) 2α L(j i) ad E(S j S i )S = (/2)E(Sj i 2 + S2 S j+i 2 ). Therefore ( σij 2 := E S j S i j i ) 2 S a( j i )(j i) 2α L(j i) a j i ( j i ) 2α L() +a j i ( j + i)2α L( j + i). Note that the fuctio f α (x) = ( x)x 2α x( x) + x( x) 2α, 0 x, has maximum value α = 2 2α 2 2 at x = /2. For give 0 < ε < /2, there exists δ > 0 such that sup 0 x ε f α δ (x) α. (2.7) By the defiitio of a slowly varyig fuctio, for 0 < ε < /2, max ε j i L(j i)/l(), as. Hece max i<j,j i ε σ 2 ij /σ2 sup ε x f α (x) = α. Moreover by property (2.4), lettig ε > 0 be small eough, for large eough we have max i<j,j i<ε σ2 ij/σ 2 max i<j,j i<ε by (2.7). Therefore ( j i )(j i )2(α δ) ( j i )( j i ) +( j i )( j i )2(α δ) α, max i<j σ2 ij /σ2 α. (2.8) For ay positive itegers i, ad M, with M < /2, let i M = [i/[/m]][/m], where [ ] deotes the largest iteger part. Note that 0 i i M [/m]. Write R() max S jm S im j M i M S + 2 max i<j S j j j M + M S. (2.9) By (2.8), we have that for large, P max S j i<j M S im j M i M S /σ ( + ε 2 ) α x M 2 exp ( + ε)x 2 /2. (2.0)

6 824 ZHENGYAN LIN Moreover, by Lemma 2. ad takig M to be large eough, we have P 2 max S j j j M /σ ε α x 4 M P max S j k[/m] /σ ε α x k[/m] j (k+)[/m] 8 k=0 CM exp ε 2 α x 2 2α L() 2 28(/M) 2α L(/M) CM exp( x 2 ) (2.) for some C > 0, provided is large eough. It is clear that for large M P M S /σ ε α x 4 exp ε2 α M 2 x 2 exp( x 2 ). (2.2) 32 Combiig (2.9) (2.2) yields PR()/σ ( + ε) α x C exp ( + ε)x 2 /2 with C = M 2 + C +. The lemma is proved. We eed the well-kow Slepia lemma. Lemma 2.4. Let G(t) ad G (t) be Gausssia processes o [0, T ] for some 0 < T <, possessig cotiuous sample path fuctios with EG(t) = EG (t) = 0, EG 2 (t) = EG 2 (t) =, ad let ρ(s, t) ad ρ (s, t) be their respective covariace fuctios. Suppose that ρ(s, t) ρ (s, t) for s, t [0, T ]. The for ay x, P sup 0 t T G(t) x P sup 0 t T G (t) x. Proof of Theorem 2.. We first prove (2.). Let θ >, k = [θ k ] ad L k = mi (k ) k L(). For (k ) < k, write R(k ) R() max 2 S j j (k ) j<k k S k + Sk S j ad hece 2σ 2 3 max S (k ) j<k k S j + 2( (k ) /k ) S k, (2.3) log log R() 2(k ) 2α L k log log(k ) R(k ) + 3 max S (k ) j<k k S j (k ) +2 ( k ) S k. (2.4) Usig the property of a slowly varyig fuctio we have L k L(k ), which implies (k ) 2α L k /σ 2 k θ 2α as k. (2.5)

7 STRONG LAWS OF R/S STATISTICS 825 The, usig Lemma 2.3 ad takig θ to be ear eough to oe, we obtai P R(k )/σ k ( + ε 2 ) (k ) 2α L k /σk 2 2α log log(k ) P R(k )/σ k ( + ε 2α 4 ) log log(k ) C exp ( + ε 4 ) log log(k ). (2.6) Similarly to (2.5), (k ) 2α L k /σk 2 (k ) (θ ) 2α. The, usig Lemma 2., we obtai S k S j P 3 max ε (k ) 2α L k (k ) j<k σ k (k ) 4 σk 2 2α log log(k ) (k ) C exp 2 log log(k ) (2.7) for some C > 0, provided θ is ear eough to oe. Similarly (k ) P 2( k ) S k ε (k ) 2α L k σ k 4 σk 2 2α log log(k ) C exp 2 log log(k ). (2.8) Combiig (2.6) (2.8) with (2.4) yields P R() ( + ε) 2α σ 2 log log (C + 2C) exp ( + ε 4 ) log log(k ) = (C + 2C)((k ) log θ) (+ε/4), which implies (2.) by the Borel-Catelli lemma. I order to prove (2.3), it is eough to show lim sup 2σ 2 log log R() α a.s.. uder (2.2). This is a cosequece of lim sup 2σ 2 2 log log 2 S 2 S α a.s.. (2.9) We have E((/2)S 2 S ) 2 = (/4)ES ES2 ES 2 S = (/4)ES ES2 (/4)(2) α L(2) + α L(), ad hece E((/2)S 2 S ) 2 /σ 2 2 (/4) + (/2) α = α.

8 826 ZHENGYAN LIN For m <, we have ( Sm 2E S ) = E(S2 m + S2 (S S m ) 2 ) = σ m + σ σ m σ σ m σ σ mα L(m) /2 α L() /2 + α L() /2 m α L(m) /2 ( α L( m) ( + m α L() /2 L(m) /2 =: p (m, ) + p 2 (m, ) + p 3 (m, ). σ2 m σ m σ m σ ) L( m) L() ( m)2α 2α ) Let θ >0 be a large iteger ad j =θ j. By (2.4), for i<j, we have p ( ki, kj )= θ k(j i)α L(θ ki ) /2 /L(θ kj ) /2 0 as k. Usig coditio (2.2), we have p 2 ( ki, kj ) 0 as k. Moreover p 3 ( ki, kj ) 2αθ k(j i)( α) [(L(θkj θ ki )) /(L(θ kj )L(θ ki ))] 0 as k. Hece E( S2ki σ 2ki Clearly from (2.20) we have that, as k, E( Ski σ 2ki S2 ) kj 0 as k. (2.20) σ 2kj S2 ) kj S2ki 0, E( σ 2kj σ 2ki S kj σ 2kj ) 0. (2.2) Let Z i = ((/2)S 2i S i )/σ 2i ad r ij = EZ i Z j. From (2.20) ad (2.2), for ay give δ > 0, we have r ki,kj δ provided k is large eough. Now let ξ j, j ad η be idepedet ormal radom variables with meas zero, Eξ 2 j = EZ 2 kj δ ad Eη2 = δ. Defie ζ j = ξ j + η. The Eζ 2 j = EZ 2 kj ad EZ ki Z kj Eζ i ζ j for i j. Let A i = Z i / 2α log log i 3ε. For small ε > 0, takig iteger k large eough ad δ > 0 small eough, with Slepia s lemma we have M M ζ P A ki P ( i i=m i=m 2α 3ε) log log ki M ξ i P ( i= 2α 2ε) + Pη ε 2α log log km log log ki M ( PN(0, ) ( ε) ) 2 log log ki i= +P N(0, ) ε 2α log log km δ M exp (log ki ) ( ε) + exp 2 log log km. (2.22) i=

9 STRONG LAWS OF R/S STATISTICS 827 Note that i= (log ki ) ( ε) =. (2.22) implies that, for km N, P A i P A ki exp 2 log log km. i=n i=m Lettig k, we obtai P i=n A i = 0, ad hece P N= i=n A i = 0, which implies (2.9). This proves Theorem 2.. Now we tur to the proof of Theorem 2.2. To this ed we eed the wellkow Borell iequality. Lemma 2.5. Let X t, t T be a cetered separable Gaussia process with almost surely bouded sample paths. Let X = sup t T X t. The for all x > 0, where σ 2 T = sup t T EX 2 t. P X E X > x 2 exp x 2 /(2σ 2 T ), Proof of Theorem 2.2. From Theorem 2., to show Theorem 2.2 it suffices to prove that 0 for α (/2, ), S 2 ()/σ 2 (2.23) for α (0, /2). Usig Lemma 2.2, (2.23) is equivalet to 0 for α (/2, ), Xj 2 /σ 2 for α (0, /2). (2.24) It is well-kow that ( Xj 2)/2 = sup a j X j, where sup = sup (a,...,a ): i= a2 i. By Borell s iequality, we have that for ay x > 0, P ( Xj 2 ) /2 E( Xj 2 ) /2 x = P sup a j X j E sup a j X j x x 2 2 exp 2 sup E( a j X j ) 2. (2.25) Puttig σ 2 = EXj 2, we have sup E( a j X j ) 2 = σ sup a i a j EX i X j i<j σ sup a i a j (j i) 2α 2 L(j i). (2.26) i<j

10 828 ZHENGYAN LIN Obviously, there is a costat c 0 such that for ay i < j, (j i) 2α 2 L(j i) c 0. Put = [ ] ad L () = max <j L(j). Clearly, L () c ( α)/2 for some c > 0. Therefore sup a i a j (j i) 2α 2 L(j i) i<j c 0 sup i+ i= j=i+ a i a j + sup α L () i= j=i+ + a i a j =: β + β 2, (2.27) where, usig Cauchy-Schwarz s iequality, β c 0 sup a i /2 i+ ( a 2 j) /2 c 0 /2 /2 i= j=i+ ad similarly, sup ( i= a 2 i ) /2 c 0 3/4, (2.28) β 2 α L () c (+α)/2. (2.29) Combiig (2.26) (2.29) with (2.25) shows that for ay ε > 0, P ( Xj 2 ) /2 E( = 2 exp = Xj 2 ) /2 ε ε 2 2c 0 /4 + 2c (α )/2 <, which, i combiatio with the Borel-Catelli lemma, implies that as, ( Xj 2 ) /2 E( Xj 2 ) /2 0 a.s.. Hece, otig that 0 < b := (/) E X j ((/) EX 2 j )/2 = (EX 2 )/2 = σ, we have b 2 lim if (/) X 2 j lim sup (/) X 2 j σ2 a.s., which yields (2.24) by recallig (2.6). This completes the proof of Theorem 2.2. Refereces Billigsley, P. (986). Probability ad Measure. 2d editio. New York: Willey. Bera, J. (992). Statistical methods for data with log-rage depedece. Statist. Sci. 7, Bera, J. (994). Statistics for Log-Memory Processes. Chapma ad Hall, New York. Bhattacharya, R ad Waymire, E. (990). Stochastic Processes with Applicatios. Wiley, New York.

11 STRONG LAWS OF R/S STATISTICS 829 Borell, C. (975). The Bru-Mikowski iequality i Gauss space. Ivet. Math. 30, Cox, D. R. (984). Log-rage depedece: a review. I Statistics: A Appraisal Proceedigs 50th Aiversary Coferece (Edited by H. A. David ad H. T. David), The Iowa State Uiversity Press. Csáki, E., Csörgő, M. ad Shao, Q. M. (992). Ferique type iequalities ad moduli of cotiuity for l 2 -valued Orstei-Uhlebeck processes. A. Ist. Heri. Poicaré Probab. Statist. 28, Feller, W. (95). The asymptotic distributio of the rage of sums of idepedet radom variables. A. Math. Statist. 22, Grager, C. ad Joyeux, R. (980). A itroductio to log-rage time series models ad fractioal differecig. J. Time Ser. Aal., Hoskig, J. (98). Fractioal differecig. Biometrika 68, Hurst, H. (95). Log term storage capacity of reservoirs. Tras. Amer. Soc. Civil Egieers, 6, Li, Z. (200). A law of iterated logarithm for R/S statistics. Submitted. Li, Z. ad Lee, S. (2002). The law of iterated logarithm of rescaled rage for AR() model. Submitted. Lo, A. (99). Log-term memory i stock market prices. Ecoometrica 59, Madelbrot, B ad Va Ness, J. W. (968). Fractioal Browia motio, fractioal oises ad applicatios. SIAM Rev. 0, Mora, P. A. P. (964). O the rage of cumulative sums. A. Ist. Statist. Math. 6, 09. Seeta, E. (976). Fuctios of Regular Variatio. Lect. Notes i Math. 508, Spriger-Verlag, New York. Serflig, R. J. (970). Covergece properties of S uder momet restrictios. A. Math. Statist. 4, Slepia, D. (962). The oe-sided barrier problem for Gaussia oise. Bell. System Tech. J. 4, Taqqu, M. (975). Weak covergece to fractioal Browia motio ad to the Roseblatt processes. Z. Wahrsch. Verw. Gebiete. 3, Williger, W., Taqqu, M. S. ad Teverovsky, V. (999). Stock market prices ad log-rage depedece. Fia. Stochastics 3, -5. Departmet of Mathematics, Zhejiag Uiversity, Xixi Campus, Hagzhou, Zhejiag 30028, Chia. zli@mail.hz.zj.c (Received July 2002; accepted August 2003)