Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics, Woc aw Uivesity of Techology Abstact: I this pape, motivated by [2], we deive ecessay ad suciet coditios fo bouded ad peiodically coelated solutios to the system of equatios descibed by ARMA(,q) model. Mailig addess: The Hugo Steihaus Cete Woc aw Uivesity of Techology Wybze e Wyspia skiego 27 50-370 Woc aw, Polad (+48-7 ) 320-3530 (+48-7 ) 320-2654 hugo@im.pw.woc.pl
O ARMA(,q) models with bouded ad peiodically coelated solutios A. Weo ad A. Wy lomańska (Woc law) Abstact I this pape, motivated by [2], we deive ecessay ad sufficiet coditios fo bouded ad peiodically coelated solutios to the system of equatios descibed by ARMA(,q) model. 2000 Mathematics Subject Classificatio: 35C0, 35B35. Keywods: Peiodically coelated, ARMA model, peiodic coefficiets. Itoductio We coside a system ARMA(,q) give by the fomula: whee X b X a ξ + a ξ +... + a (q ) ξ (q ), () (X ) is a sequece of complex adom vaiables with mea 0 ad fiite vaiace i the space with the ie poduct (.,.) ad M X sp{x k : k Z}, (b ) ad (a ) ae sequeces of o zeo complex umbes, (ξ ) is a sequece of ucoelated complex adom vaiables with mea 0 ad vaiace ad M ξ sp{ξ k : k Z}. I a ecet pape H. Hud, A. Makago ad A. G. Miamee [2] gave ecessay ad sufficiet coditios fo boudedess i the geeal case of AR() model ad the specifically fo peiodic ad almost peiodic coefficiets (a ). The peset effot is a attempt to udestad the situatio i a impotat fo applicatios case of ARMA(,q) models [,3]. Such systems aise i climatology, ecoomics, hydology, electical egieeig ad othe disciplies. I Sectio 2 we discuss the elatioship betwee existece of bouded solutios to the system equatios descibed by ARMA(,q) model ad coditios o thei coefficiets (Theoem ). Next, peiodically coelated solutios ae examied (Theoem 2). I Sectio 3 we simplify the cosideatio fo q 2. The fial Example povides a egative aswe to the questio: Whethe o ot fo the fact that system () has a PC solutio it follows that the sequeces of coefficiets (b ) ad (a ) ae peiodic? This gives a patial solutio to the questio (i the case of AR() system) studied i [2]. Let us deote: s B s with the covetio that B s if > s. It is easy to show, that iteatig k-times the equatio () we obtai: X +k B +k + X + k j b j q B +k +j+ ξ +j s a +j s, (2)
X k X B k+ k 2 The ARMA(,q) model B k+j k+ q ξ k+j s a k+j s. (3) Defiitio A stochastic sequece is called bouded if sup X. Lemma If sup B ad system () has a bouded solutio i Mξ, the: q q k0 a +j k 2 B +j + B+j (k s) + ] <. (4) PROOF: If sup B the thee exist subsequece k of positive iteges such that lim B k. So we have fo all Z: lim B +k +. If system () has a bouded solutio the fom (2) we obtai: Hece: We obtai: k X + B +k q +j+ B +k + ξ +j s a +j s X +k B +k + k q X lim[ B +j ξ +j s a +j s ]. + k q B +j ξ +j s a +j s 2 + k q q k0 Because X is a bouded solutio of system (), so we obtai: 0. a +j k 2 B +j + B+j (k s) + lim k q B +j + ξ +j s a +j s 2 ] sup X 2 <, hece q q k0 a +j k 2 B +j + B+j (k s) + ] <. Lemma 2 If sup B 0 ad system () has a bouded solutio i Mξ, the: q q k0 a j k 2 B j+b j+ (k s) ] <. (5) 2
PROOF: If sup B 0, the thee is a subsequece (k ) of positive iteges such that: lim B 0 k. Fo all Z we have: lim B +k. Because system () has the bouded solutio we have fom (3): k q X B k ++jξ k+j sa k+j s X k B k + 0, So we obtai: k q X lim[ B k ++jξ k+j sa k+j s] 0 q lim[ B + jξ j s a j s ]. j k + Sice X is the bouded solutio of system () ad ξ is the othoomal basis i Mξ hece we have: 0 q X 2 lim B + jξ j s a j s 2 j k + We obtai the: q q a j k 2 B j+b j+ (k s). k0 q q k0 a j k 2 B j+b j+ (k s) ] sup X 2 <. If sup B ad sup B 0, the system () has a bouded solutio. But thee is a thid possible coditio, which gives a bouded solutio of (): sup B < ad sup B 0 <. (6) Lemma 3 If coditio (6) holds ad system () has a bouded solutio, the: ad k k k q w0 s k q q w0 q a j w 2 Bj+B k j w+s k ] < (7) a k+j w 2 B k+j k+ B k+j w+s k+ ] <. (8) 3
PROOF: We use (2) ad (3) (ad povide 0). We assume fo all k Z ad some C that we have B k < C ad B 0 k < C. Fo all k > 0 we the have: k k k q q w0 s k q q w0 a j w 2 B k j+b k j w+s ] sup k a k+j w 2 B k+j k+ B k+j w+s k+ The solutio of system () is give by: B k X + Bk j+ X k 0 jk+ X B k X 0 2 sup X k 2 ( + C) 2 <, k ] sup X k X 0 k B k+ 0 q ξ j sa j s if k > 0, X if k0, X B 0 k+ B j k+ q ξ j sa j s if k < 0, 2 sup X k 2 ( + C) 2 <. k (9) whee X is a adom vaiable i Mξ. Theoem System () has a bouded solutio if ad oly if oe of the followig holds: (I) sup B ad (II) sup B 0 ad q q k0 a +j k 2 B +j + B+j (k s) + ] < q q a j k 2 B j+b j+ (k s) ] <, k0 (III) sup B <, sup B 0 < ad k k k q w0 s k q q w0 q a j w 2 Bj+B k j w+s k ] < a k+j w 2 B k+j k+ B k+j w+s k+ ] <. PROOF: If coditio (I) holds, the the solutio of system () give by the followig fomula: X is bouded. If coditio (II) holds, the X defied by: X q B +j ξ +j s a +j s (0) + B+ j q ξ j s a j s () is the bouded solutio of system (). If coditio (III) holds, the X give by fomula (9) is bouded ad is a solutio of system (). I lemmas, 2 ad 3 it is show that if X is a bouded solutio of system (), the oe of coditios (I), (II) o (III) holds. 4
Defiitio 2 A stochastic sequece (X ) is called peiodically coelated (PC) with peiod T if fo all k sequece (X +k, X ) is peiodic i with peiod T, i.e, (X +k, X ) (X +k+t, X +T ). Theoem 2 If (b ) ad (a ) ae peiodic with the same peiod T ad P b b 2...b T, the system () has a bouded solutio if ad oly if P. Moeove, the solutio is PC with the same peiod T ad: (i) If P >, the the solutio is give by (0). (ii) If P <, the the solutio is give by (). PROOF: (i) If P >, the fo all Z we have: q q k0 a +j k 2 B +j + B+j (k s) + T q q N0 w k0 +NT +w B+ B a +NT +w k 2 +NT +w (k s) + N0 P 2N T q q w k0 a +w k 2 B+ +w B+w (k s) + P 2 T q q w k0 a +w k 2 B +w + B+w (k s) + Theefoe (4) holds ad X defied by (0) is the bouded solutio of system (). (ii) If P <, the fo all Z we obtai: p q a j k 2 B j+b j+ (k s) k0 T q q a NT w k 2 B NT w+b NT w+ (k s) N0 w k0 T q q P 2N a w k 2 B w+b w+ (k s) N0 P 2 w k0 T q q a w k 2 B w+b w+ (k s) <. w k0 Theefoe (5) holds. X defied by fomula () is bouded ad satisfies fomula (). <. I the ext sectio it is show fo q 2 that (X ) defied by fomulas (0) o () is peiodically coelated ad the coditio P violates the coditios (I), (II) ad (III) of Theoem. Theefoe, system () has o bouded solutio if P. 5
3 The ARMA(,2) model Fo simplicity of otatio we coside hee oly the ARMA (,2) case: X b X a ξ + a ξ. (2) Theoem 3 If (b ) ad (a ) ae peiodic with the same peiod T ad P b b 2...b T, the system (2) has a bouded solutio if ad oly if P. Moeove, the solutio is PC with the same peiod T ad is give by (0) if P > ad is give by () if P <. PROOF: We will split the poof i 3 cases. (i) I view of Theoem 2 we have that if P >, the fo all Z coditio (4) holds. Hece thee is a bouded solutio of (2). The solutio is give by fomula (0) fo q 2. Now we wat to show, that the stochastic sequeces (X ) i fomula (0) ae PC with peiod T. We take ay k, ad we have: B +k + (X +k, X ) (3) a +k+j 2 (+ 2 + + )+ a +k 2 ( + 2 ). b +k+j+ b +k+j+ b +k+j+ b +k+ b +k+ B +k+j +k+ B+ +k The coelatio fuctio is bouded ad sice coefficiets (b ) ad (a ) ae peiodic with peiod T, theefoe fom (3) we obtai: (X +k, X ) (X +T +k, X +T ). (ii) Similaly, fom Theoem 2 we have that if P <, the fo all Z coditio (5) holds. Hece thee is a bouded solutio of (2). The solutio is give by fomula () fo q 2. The coelatio fuctio is give by: B +k + (X +k, X ) (4) a +k j B +k +k j+2 2 (+ b +k j+ 2 +b +k j+ +b +k j+ )+ a +k 2 ( b +k 2 +b +k ). j2 B +k + Because the coelatio fuctio is bouded ad (b ) ad (a ) ae peiodic with peiod T, theefoe fom (4) we obtai: (X +k, X ) (X +T +k, X +T ). Thus by the above coditios (X ) is PC with peiod T. (iii) If P, the a +j 2 ( + 2 + b +j+ B +j + b +j+ + ) + a 2 ( + ) b +j+ b + b + ad a j B j+2 2 ( + b j 2 + b j + b j ) + a 2 ( b 2 + b ) j2 6
which violates coditios (I) ad (II) of Theoem. Sice the N k k k NT NT + Bj+2 a j 2 ( + b j+ 2 + b j+ + b j+ ) NT + BkT +2 a kt 2 ( + b kt + 2 + b kt + + b kt + ) P 2 N a 0 2 ( + b 2 + b + b ) k a 0 2 ( + b 2 + b + b ) N P 2 Bj+2a k j 2 ( + b j+ 2 + b j+ + b j+ ) + a 0 B2 k 2 (b + ) + a k 2 (b k+ + b k+ 2 )] k sup [ knt + Bj+2a k j 2 (+ b j+ 2 +b j+ +b j+ )+ a 0 B2 k 2 (b +)+ a k 2 (b k+ + b k+ 2 )] which violates coditio (III) of Theoem. Theefoe, i view of Theoem, system (2) has o bouded solutio if P. Fially, we show that thee exists a ARMA(,2) system with bouded ad PC solutio fo which coefficiets (a ) ae ot peiodic. Example Let us coside the system ARMA(,2) give by: We have: The coefficiets fulfill coditio (4): a +j 2 B +j ( + 2 + + 2 b +j+ X + 2X ( 2) ξ + ( 2) ξ b 2, a ( 2). b +j+ + ) + a 2 ( + ) b +j+ b + b + 2 +j 4 j ( 2 )2 2 ( 2 ) 0. Theefoe by Theoem the solutio of the system is give by fomula (0). The coelatio (X +k, X ) is give by: (X +k, X ) a +k+j 2 B+ +k B +k+j (+ 2 + + )+ a +k 2 +k+ 2 b +k+j+ b +k+j+ b +k+j+ B+ +k ( + 2 ). b +k+ b +k+ So we have: (X +k, X ) ( )k 2 k 2 +k+j 4 j+ ( ) k 2+k 42 k ( )k 2 ( )k 2 0. 4 4 Theefoe (X ) is PC (as the coelatio fuctio is costat), but the coefficiets (a ) ae ot peiodic. Ackowledgmets The wok was suppoted i pat by the KBN gat P BZ KBN 06/P 03/99. 7
Refeeces [] J.D.Hamilto, 994. Time Seies Aalysis, Piceto Uivesity Pess, Piceto. [2] H.Hud, A.Makago, A.G.Miamee, 2002. O AR() models with peiodic ad almost peiodic coefficiets, Stoch. Poc. Appl.,00, 67-85. [3] M.Pouahmadi, 200. Foudatios of Time Seies Aalysis ad Pedictio Theoy, Willey, New Yok. Hugo Steihaus Cete Istitute of Mathematics Woclaw Uivesity of Techology 50-370 Woclaw, Polad E-mail adesses: weo@im.pw.woc.pl, wyloma@im.pw.woc.pl 8