On processes with hyperbolically decaying autocorrelations

Transcription

1 O processes with hyperbolically decayig autocorrelatios Łukasz Dębowski Istitute of Computer Sciece, Polish Academy of Scieces ul. Ordoa 1, Warszawa, Polad Abstract We discuss some relatios betwee autocorrelatios (ACF) ad partial autocorrelatios (PACF) of weakly statioary processes. Firstly, we costruct a extesio of a process ARIMA(0, d, 0) for d (, 0), which ejoys osummable partial autocorrelatios ad autocorrelatios decayig as rapidly as ρ 1+d. Such a situatio is impossible if the absolute sum of autocorrelatios is sufficietly small. We show that the the PACF is less tha the ACF up to a multiplicative costat. Our secod result complemets a similar result of Baxter (196). Key words: autocorrelatio, partial autocorrelatio, ARIMA processes The work was partially supported by the Polish Miistry of Scietific Research ad Iformatio Techology, grat o. 1/P03A/045/8. 1

2 1 Itroductio Deote by (ρ k ) k Z the autocorrelatio fuctio (ACF) of a zero-mea complexvalued weakly statioary process (X i ) i Z. For simplicity of the followig formulae, let us put Var X i = 1. The, ρ i j = Cov(Xi, X j), where Xi stads for the complex cojugate of X i. O the other had, itroduce the partial autocorrelatio fuctio (PACF) ( [Xi ] ) α i j = Corr P {j,j+1,...,i 1} X i, Xj P {j,j+1,...,i 1} X j, where P K X j = j K φk ij X j is the best liear predictor of radom variable X i i terms of variables X j, j K. I a disguised form, the PACF appears i the theory of orthogoal polyomials o the uit circle (OPUC) uder the ame of Verblusky coefficiets or Schur parameters (Simo, 005, Sectio 1.1). I this article, we shall study certai liks betwee the ACF ad the PACF. To place our results i cotext, let us first recall a few kow facts. Although the defiitio of the PACF seems more ivolved tha that of the ACF, the parameterizatio of the process i terms of (α k ) k N is ucostraied, i.e., the sole coditio o α k reads α k 1 for k 1 ad α = 1 = α m = 0 for m > 1 (1) (Schur, 1917; Ramsey, 1974). I cotrast, the autocorrelatio matrix (ρ i j ) i,j= has to be oegative defiite. From this perspective it appears more plausible to characterize time series i terms of their PACF rather tha the ACF. This opiio has bee also advocated by Ioue (008). I fact, othig is lost by such reparameterizatio sice there exists a oe-to-oe correspodece betwee the fiite sequeces (ρ i ) i=1 ad (α i ) i=1. For example, the ACF ca be computed recursively from the PACF via the Szegő-Durbi-Leviso recursio ad a special case of the Yule-Walker equatios φ k = φ 1,k α φ 1, k () φ iρ i = 0, (3) i=0 where 0 k, φ 0 :1, φ,+1 := 0, ad φ i := φ {1,,...,} +1,+1 i otherwise (cf. Brockwell ad Davis, 1987, Propositio 5..1). I the preset paper, we will research processes with a hyperbolically decayig ACF. As we will show, the PACF may be summable or osummable for such processes, depedig o the value of the absolute sum of autocorrelatios. If that sum is sufficietly small the the PACF mimics the behavior of the ACF. Whe the sum is too large the we may have a osummable PACF, eve if the autocorrelatios decay asymptotically very fast. The existece of processes with a osummable PACF ad rapidly decayig ACF ca be hypothesized i view of the formula k= (±1) k ρ k = k= (±1) k α k 1 (±1) k α k. (4)

3 This expressio was derived by Dębowski (007) oly for real-valued processes which satisfy α k < ad α k < 1 for all k 1 (5) but we suppose that relatioship (4) holds i a more geeral case. Namely, we cojecture that { if f(0) = ρ k = α k = ad α k > 0, 0 if α k = ad α k < 0 k= holds for spectral desity f(ω) = k= ρ ke ikω. Processes with f(0) = are kow as processes with log memory, while those with f(0) = 0 are called atipersistet (Bera, 1994). Let us ote that the PACF is absolutely osummable or satisfies α k = 1 for some k whe f(0) = or f(0) = 0, accordig to the Baxter (1961, 196) theorem. That theorem claims that coditio (5) is equivalet to k= ρ k < ad f(ω) > A for a certai A > 0. (6) This result suggests that certai examples of processes with a absolutely osummable PACF ad absolutely summable autocorrelatios may be searched amog atipersistet processes. The orgaizatio of this paper is as follows. I Sectio, we will demostrate that the defiitio of processes ARIMA(0, d, 0), itroduced by Hoskig (1981) for d ( 1/, 1/), ca be exteded to d (, 0). This yields ideed atipersistet processes with a egative osummable PACF ad autocorrelatios decayig as rapidly as 1+d. O the other had, if the autocorrelatios have the same asymptotics but their sum is sufficietly small the the PACF must be absolutely summable. I Sectio 3, we will show that α k AG(k) if ρ k G(k) for a sufficietly fast decayig fuctio G. Fuctio G may have a power-law tail but it is required that G(k) be small eough. This result complemets a boud for the momets of PACF ad ACF derived by Baxter (196) for processes that satisfy coditio (6). Processes ARIMA(0, d, 0) with d (, 1/) Hoskig (1981) costructed processes ARIMA(0, d, 0) with d ( 1/, 1/) which exhibit ρ 1+d ad α 1. The autocorrelatios of these processes are absolutely osummable for d > 0 ad summable to 0 for d < 0. Whereas Bodo ad Palma (007) observed that the defiitio of ARIMA ca be exteded to d > 1, below we complete the extesio to d 1. Theorem 1 For d (, 1/) there exists a weakly statioary process, kow as ARIMA(0, d, 0) i the case of d ( 1/, 1/), that has parameters i + d 1 ρ = i d, (7) i=1 ( ) (k d 1)!( d k)! φ k, (8) k ( d 1)!( d)! α = d d, (9)

4 where z! := Γ(z + 1). Moreover, as for the asymptotics of the ACF, we have lim ρ / 1+d = ( d)!/(d 1)! if d + 1 N. We also have k= ρ k = for d (0, 1/) ad k= ρ k = 0 for d < 0. Proof: Observe that (α ) N defied i (9) satisfies α 1 for 1 ad d (, 1/) ad therefore is a PACF of a weakly statioary process. The partial autocorrelatios determie the coefficiets φ k ad ρ k through iteratios () ad (3) uiquely give the iitial coditios φ 0 1, φ,+1 = 0, ad ρ 0 = 1. These iitial coditios are clearly satisfied. Hece to demostrate that (8) ad (7) are the appropriate coefficiets pertaiig to the process, it suffices to check that () is satisfied for 1 ad 0 k give (8) (9) ad that (3) is satisfied for 1 give (7) (8). Ideed, for (8) ad (9) we obtai (): φ 1,k d d φ 1, k [( ) k d ( ) ] k (k d 1)!( 1 d k)! k d k ( d 1)!( 1 d)! ( ) k d (k d 1)!( 1 d k)! k d ( d 1)!( 1 d)! ( ) (k d 1)!( d k)! = φ k. k ( d 1)!( d)! O the other had, for (7) ad (8) we obtai (3): ( ) k k φ i=1 (i d 1) i=1 (i + d 1) kρ k k i=1 (i d) [( )] 1 d ( )( ) k d 1 k + d 1 k k [( )] 1 d ( )( ) d d ( 1) k k [( d )] 1 ( d d ) ( 1) = 0, where we used the upper egatio formula ( ) r ( ) k = ( 1) k k r 1 k ad the Cauchy formula ( r )( s ) ( k k = r+s ). Both formulae hold for all r, s R (Graham et al., 1994, Chapter 5, Table 0). I the sequel, let us establish the asymptotics for the autocorrelatios. For d + 1 N we have ρ = ( d)!( + d 1)!/( d)!(d 1)!. By the Stirlig approximatio lim z Γ(z)[e z z z 1/ π] 1 = 1 we have lim x Γ()/Γ(+ x) = 1 so lim ρ / 1+d = ( d)!/(d 1)!. Fially, we ispect the sum of autocorrelatios. Notice that k= ρ k = i=1 (i + d)/(i d) follows by iductio o. Hece k= ρ k = for d > 0 ad k= ρ k = 0 for d < 0. 3

5 3 A boud for idividual PACF values I the previous sectio, we have show that the PACF may decay slower tha the ACF if the partial autocorrelatios do ot satisfy coditio (5). This is impossible i the other case. If (5) holds the k=1 km α k < follows from k=1 km ρ k < by Theorem.3 of Baxter (196), for ay m N. Baxter origially proved that implicatio uder the assumptio of a positive ad cotiuous spectral desity but, by the equivalece of coditios (5) ad (6), the spectral desity is positive ad cotiuous if the process satisfies (5). While the Theorem.3 by Baxter (196) bouds the momets of ACF i terms of the momets of PACF, we will additioally derive a iequality for idividual ρ k ad α k for certai processes that satisfy coditio (5). Cosider a fuctio G : N {0} R that obeys the followig: (i) G(0) = G(1), (ii) G is oegative ad oicreasig, (iii) there exists a B > 0 such that G( / ) BG() for N, (iv) 4B G(k) < 1. Coditios (i) (iv) are satisfied, e.g., if G() β (β 1) [max {1, }] β, β > 1. (10) β 1 Moreover, if these coditios are satisfied the there exists a statioary process such that ρ 0 = 1 ad ρ G() for 1. Namely, B must be at least 1 if G does ot grow. Cosequetly we have 4 k=1 G(k) < 1 so the Fourier trasform of the ACF (i.e., the spectral desity) is positive. By Corollary 4.3. of Brockwell ad Davis (1987), the latter coditio guaratees the existece of the requested process. I the followig, otice that α = φ. Thus we obtai α AG(). Theorem Let fuctio G obey coditios (i)-(iv). If the autocorrelatio fuctio of some weakly statioary process satisfies ρ G() for 1 the the coefficiets φ i of the best liear predictors obey for 1 i ad some costat A. φ i AG(i) (11) Proof: Notice that the geeral Yule-Walker equatios φ K ij = ρ j i φ K ikρ j k, j K. k K\{j} have exactly oe solutio for (φ K ij ) j K. We ca therefore express φ K ij = ρ K (i; j), (1) =1 4

6 where ρ K (i; j) = { ρ j i if = 1, k K\{j} ρk 1(i; k)ρ j k if, provided the series =1 ρk (i; j) coverges for each j K. Now let us show that this covergece actually happes uder the assumed hypothesis. By the way, we will obtai the requested iequality for φ K ij. Write first 1 {i=j} if = 0, G (i; j) = G( i j ) if = 1, k Z G 1(i; k)g( k j ) if, where 1 { } deotes the idicator fuctio. I the sequel, iequalities ρ k G(k) for k 1 imply ρ K (k 0 ; k ) k 1 K\{k } k 1 Z k 1 Z k 1 K\{k } j=1 ρ kj k j 1 [ 1{k1=k 0} + G( k 1 k 0 ) ] G( k j k j 1 ) j= = G 1 (k 0 ; k ) + G (k 0 ; k ), (13) To boud G (i; j), observe ow that for i j we have G( i k )G( k j ) k Z k i+j [ 4 ( j i G( i k )G ) + k i+j ( j i G ) G( k j ) ] ( ) j i G(k) G CG( i j ), (14) where C = 4B G(k). Of course, the same iequality holds for i j. By a iterated applicatio of (14) we obtai G (i; j) C 1 G( i j ) for 1. Combiig that with (1) ad (13) yields φ K ij ρ K (i; j) 1 {i=j} + G( i j ). (15) 1 C =1 By φ i = φ {1,,...,} +1,+1 i ad (15), iequality (11) follows for A = (1 C) 1. Ackowledgmet The author wishes to thak Ja Mieliczuk, Jacek Koroacki, ad a aoymous reviewer for the discussio. 5

7 Refereces Baxter, G. (1961) A covergece equivalece related to polyomials orthogoal o the uit circle. Trasactios of the America Mathematical Society 99, Baxter, G. (196) A asymptotic result for the fiite predictor. Mathematica Scadiavica 10, Bera, J. (1994) Statistics for Log-Memory Processes. New York: Chapma & Hall. Bodo, P. ad Palma W. (007) A class of atipersistet processes. Joural of Time Series Aalysis 8, Brockwell, P. J. ad Davis, R. A. (1987) Time Series: Theory ad Methods. New York: Spriger. Dębowski, Ł. (007) O processes with summable partial autocorrelatios. Statistics ad Probability Letters 77, Graham, R. L., Kuth, D. E., ad Patashik, O. (1994) Cocrete Mathematics. A Foudatio for Computer Sciece. Readig: Addiso-Wesley. Hoskig, J. R. M. (1981) Fractioal differecig. Biometrika 68, Ioue, A. (008) AR ad MA represetatio of partial autocorrelatio fuctios, with applicatios. Probability Theory ad Related Fields 140, Ramsey, F. L (1974) Characterizatio of the partial autocorrelatio fuctio. The Aals of Statistics, Schur, I. (1917) Über Potezreihe, die im Ier des Eiheitskreises beschrakt sid. Joural für die Reie ud Agewadte Mathematik 147, Simo, B. (005) Orthogoal Polyomials o the Uit Circle. Providece: America Mathematical Society. 6