A NOVEL DIFFERENCE EQUATION REPRESENTATION FOR AUTOREGRESSIVE TIME SERIES

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1 Joural of Thorcal ad Appld Iformao Tchology h Spmbr 4. Vol. 67 No. 5-4 JATIT & LLS. All rghs rsrvd. ISSN: E-ISSN: A NOVEL DIFFERENCE EQUATION REPRESENTATION FOR AUTOREGRESSIVE TIME SERIES B.SELVARAJ, M.RAJU, 3 M.THIYAGARAJAN Da, Dparm of Scc ad Humas, Nhru Isu of Egrg ad Tchology, Combaor, Taml Nadu, Ida 645. Dparm of Scc ad Humas, Nhru Isu of Egrg ad Tchology, Combaor, Taml Nadu, Ida Da Rsarch, Nhru Isu of Egrg ad Tchology, Combaor, Taml Nadu, Ida 645. E-mal: profssorslvara@gmal.com, raumurugasamy@gmal.comad 3 mhyagaraa4@gmal.com ABSTRACT Th maor compos of h m srs ar h log rm rd, h shor rm rd, cyclc varao ad rrgular flucuaos. Varous amps hav b mad o gv cssary codos for procssg h spcfc compos. Hr w ak cssary codos o prdc h asympoc bhavor of h m srs usg scod ordr dffrc of h combaos of obsrvaos obad from a gral m srs. Spcfc llusraos ar gv o auhca our clam. Kywords: Scular Trd, Tm Srs, Dffrc Equaos, Scod Ordr Forward Dffrc, Asympoc Bhavor, Numrcal Daa. Subc classfcao: 37M, 39A.. INTRODUCTION J.Nyma[]obsrvd currly h prod of dyamc drmsm scc, hr s hardly a srous pc of rsarch, whch, f rad ralscally, dos o volv opraos o sochasc procsss. Drmsc ad probablsc modls hav b sudd o ral m daa of m srs by M.G.Kdall[5] ad avrag procsss whch hlp o o prdc log m rd of h m srs. Ths cocs a lar combao of ay coscuv obsrvaos of a gv m srs. I hs papr, h modl such lar combaos as a rsul of scod ordr dffrc of ay wo lar combao of a gv srs. Ths s a oucom of a dcay sudy of h asympoc bhavor of squc udr suabl combao of scod ordr fucoal dffrc. Ths s a so of h paprs sudd by M- RogXu ad al.[5], ad Yu-Pg Zhao ad X- La Lu[7]. Th rlao suggsd by hs sudy pavs a way o f v ordr frs yp rcprocal quao ad hr soluos, whch plas clams mad our lmma. Spcfc llusraos o hdffrc quao suggs h vald modls for h auo rgrssv procsss. Hr w cosdr h scod ordr ural dlay dffrc quaowh w codos. R.P.Agarwal[], R.P.Agarwal ad al[]. dscussd h gral hory of dffrc quaos. May rfrcs o som applcaos of h dffrc quaos dscussd by Walr G.Klly ad Alla C.Prso[4]. Ths papr s orgazd as follows: I sco, w gv basc cocps ad rsuls. Modls o m srs ad our ovl rsuls h asympoc bhavor ar gv sco 3. Sco 4 dals wh llusrao for m srs modl ad dffrc quaos. Las sco gvs our corbuo ad fuur work hs drco.. BASIC CONCEPTS AND RESULTS W cosdr h scod ordr ural dlay dffrc quaos of h form ( + p q σ ) + f (, ) τ, () whr p,τ, q, σ, for,, N {,,,...}, l { s,..., }, s ma {τ, σ}, τ ma < {τ }, σ ma < {σ }, s h forward dffrc opraor dfd by + ad h couous fuco f: N {} R R s crasg or dcrasg ad y ad f(, y) y >, for y. W us h followg oaos hroughou, N {,,,...}, h s of aural umbrs cludg zro; N (a) {a, a +, a +,...}, whr a N. 78

2 Joural of Thorcal ad Appld Iformao Tchology h Spmbr 4. Vol. 67 No. 5-4 JATIT & LLS. All rghs rsrvd. ISSN: E-ISSN: May auhors [6,, 3, 4, 6] hav sudd h cass of p ad fs crasg, h auhor [] has sudd h cass of q. Fw auhors [6, 7] hav sudd h cass of p h frs ordr dffrc quaos. Hr w cosdr h quao () wh h followg assumpos: (C) p, <q, (C) p, q >, (C3) p >, <q, (C4) p >, q >, (C5) lmf f,, for M >. s M s Dfo.:By a soluo of quao (), w ma a ral squc { } whch s dfd for all k m k N() {τ k, σ k } ad sasfs quao () for suffcly larg k N (a), a N. A orval soluo { } of quao () s sado b ooscllaory f s hr vually posv or vually gav, ad ohrws s oscllaory.a quao s oscllaory f all s soluos ar oscllaory. Dfo.:A srs of obsrvaos (), T mad squally m cosus a m srs. Eampls of daa ak ovr a prod of m ar foud abudac dvrs flds such as mrology, gophyscs, bophyscs, coomcs, commrc, commucao grg sysms aalyss c. Daly rcords of rafall daa, prcs of a commody c. cosu m srs. Th vara dos m,.., chags occur m. Bu hs d o always b so. For ampl, h rcords of masurms of h damr of a ylo fbr alog s lgh (dsac) also gv a m srs. Hr dos lgh. Dfo.3:Th ssal fac whch dsgushs m srs daa from ohr sascal daa h spcfc ordr whch obsrvaos ar ak. Whl obsrvaos from aras ohr ha m srs ar sascally dpd, h succssv obsrvaos from a m srs may b dpd, h dpdc basd o h poso of h obsrvao h srs. Th m may b a couous or a dscr varabl. A gral mahmacal modl of h m srs Y(), T s gv as Y()f()+X().Hr f() rprss h sysmac par ad X() rprss h radom par. Ths wo compos ar also kow as sgal ad os rspcvly. Th modl s horcal: f() ad X() ar o sparaly obsrvabl. Whl h modl for Y() gvs h srucur of h grag procss, a s of obsrvaos(or m srs daa) s a ralzao of a sampl fuco of h procss. Th ffc of m may b boh h sysmac ad h radom pars. W shall us h followg propososfor modl I. Proposo.: Cosdr h dffrc quao ( + p k ) + q ma [,] s. () L z + p k. L h followg codos hold: (H) k ad l ar ogav grs, (H) {p } s a ral squc, (H3) {q } s a squc of ogav ral umbrs, (H4) s q, ad s p p p. Th h followg assros ar vald: (a) If > vually, h hr z <, z < ad z,vually ad lm z lm z or z <, z > ad z, vually ad lm z lm z. (b) If < vually, h hr z >, z > ad z,vually ad lm z lm z or z >,z < ad z,vually ad lm z lm z. Proposo.:L codos H proposo. hold ad < p p. If { }s a ooscllaory soluo of quao (), h lm. 3. ASYMPTOTIC BEHAVIOR OF DIFFERENCE EQUATION IN TIME SERIES 3.. Asympoc Bhavor of Dffrc Equao Thorm 3..: If o of h codos (C) ad (C3) s sasfd alogwh (C4), h vry ooscllaory soluo of h quao () ds ozro as. If h codo (C) s sasfd, h vry ooscllaorysoluo of quao () ds o or as. Proof: Whou loss of graly w may assum ha { } b a vually posv soluo of quao (). Th hr ss N () such ha >, for N ( ). I follows ha τ, σ ad >, for N( ),, N, whr + s, s ma {τ, σ}. 79

3 Joural of Thorcal ad Appld Iformao Tchology h Spmbr 4. Vol. 67 No. 5-4 JATIT & LLS. All rghs rsrvd. ISSN: E-ISSN: L z + p q, for N ( ). τ σ (3) Frs w cosdr h codo (C). Th from quao (3), w hav z q, for N ( ). (4) σ I follows from quao (.) ha (, ) < z f, for N ( ). (5) By proposo., w hav z <. W clam lm. Cas(): Suppos fs crasg. Th by proposo., w aa our clam. Cas() Suppos f> s dcrasg. W assum ha lm. Th hr ss af subsquc { () } {} such ha lm ( ) >. Thrfor w ca fd a squc of subss N () N such ha () N(), for N (). So, hr ss N () such ha >, for N ( ).Thus f (, ) < f,, for N ( ). (6) Now, quals (5) ad (6) mpls ha z >, f. Summg h abov qualy from M N ( ) o, w oba z > f s,. s M Aga summg from M N ( ) o, w hav z > f,. s M s By h codo (C5), w s ha z > as. Ths s a coradco o h proposo.. Thus lm. N, w cosdr h codo (C3). I hs cas z s quao (3) ad cosquly from quao (), w hav (, ) < z f, for N ( ). To prov z <, for N ( ). Suppos ha z, for N ( ). Th hr ss 3 N ( ) ad k > such ha z k. Thrfor, fromquao (3), w hav k p q, for N ( 3 ). (7) τ + σ Now wo cass ars. Suppos { } s uboudd..., lmsup. Th hr ss a subsquc { } u u N such ha u 3 +σ ad u as u ad u l { } ma. I vw of h qualy(7), w u l hav k p q. u u u τ + k + p u σ q p u τ u σ. u p, whch lads o a coradco. Suppos { } s boudd,.., lmsup <. Th hr ss a subsquc { } u u ha u ad l { } u Nsuch as u ad ma. Th, w hav lm sup. u u I vw of h qualy (7), w hav k p + q. u u τ u σ u Takg h supror lm as u, w oba k p + q. k + p q p. p, whch also lads a coradco. I boh cass w oba hcoradco o h assumpo z. Thus z <. Th proof of hrmag s h sam as h frs par ad hc w om. Fally, w cosdr h codo (C). Th from h frs par of hproof, w s ha h quao (4), qualy (5) ad hc z < hold. Now w prov h clam lm by h followg coradco ha s mpossbl ha lm. Suppos ha lm. Th from quao (4), w oba lm z. Thrfor from quao (4), w hav q σ <, for N ( ). Now, l us df { σ } Thrfor w oba γ m. < q γ u. Takg lm as, w oba q >.Ths lads a coradco o q >. N, w shall show ha s possbl ha lm. Suppos ha lm. Th hr ss a subsquc { () } N such ha < lm <. Thus hr ss 3 N ( ) ( ) ad N () such ha () N ( 3 ) ad ( ) >, for N ( ). Now, w ma. Th from h quao df α { σ } α α < (4), w oba z q σ α. 8

4 Joural of Thorcal ad Appld Iformao Tchology h Spmbr 4. Vol. 67 No. 5-4 JATIT & LLS. All rghs rsrvd. ISSN: E-ISSN: q z α. q.ths lads o a coradco o h fac ha q. Thus boh cass,w provd ha h soluo of quao () ds o as.th smlar argums ar usd for provg h rsul whl { } s vually gav ad hc w om. Hc h horm s complly provd. Corollary 3..:Assum ha f(, v) r v β,for all v >, whr β s h rao of odd posv grs such ha < β < ad r >, for all N. Suppos ha for ay squc of subss N () N, r.i N() addo o h codo (C4), vry ooscllaory soluo { } of quao () sasfs hr lmf, or lmsup. Proof: Whou loss of graly w may assum ha { } b a vually posv soluo of quao (). Th hr ss N () such ha >, for N ( ). I follows ha τ, σ ad >, for N ( ),, N, whr + s, s ma {τ, σ}. Cosdr h codo (C4), ad h from quao (3), w hav h sam quao (3) wh h codo (C4). I follows from quao () ha h sam qualy (5) wh codo (C4).Summg h w qualy (5) from M > o, w oba z < z M rs s M α s <, for N ( ). Thr ssς > such ha z M ς, for N ( ). Aga summg h abov qualy from M > o, w oba α z < ς r <, for N ( ). s M s From h abov wo quals, w hav z < ad z < rspcvly,for N ( ). Thus lmf ad lmsup ar follows from h proposo. ad proposo.. Hc h corollary s complly provd. 3.. Auo Rgrssv Procss (AR Procss) Th procss {X()} gv by X +b X - +b X - + +b X -, b, (8) whr{ }s purly radom procss, wh ma, s calld a auorgrssv procss of ordr.x ca b obad as a soluo of h lar sochasc dffrc quao g(b)x, whr g(b) b r B r r, b. (9) z B, z z,.., Suppos ha g(b) ( ) z,..., z ar h dsc roos of h quao g(z). Furhr suppos ha z < for all,.., all h roos of g(z) l ousd h u crcl; h roos z of h characrsc quao r f ( z) b r z (whr f ( z) z g( z ) r ) all l wh h u crcl. Th compl soluo of (9) ca b wr as X r Now g ( B) Ar z r + g ( B) ( zb) r br B r r r, whr A r s ar cosas. r, b whr r ar cosas volvg z s., If h procss s cosdrd as bgu log m ago, h h corbuo of A r z r damps ou of sc.x s h gv by r r r r X. () Thus a AR procss ca b rprsd by a MA procss of f ordr. Th coffcs r of -r h rgh had sd of () ca also b obad as follows. Usg h prssos for X as gv (), for, -,,- ad h subsug (8), w g r r + b r r + b r r + b r r Equag h coffcs of, -,... from boh sds, w g. 8

5 Joural of Thorcal ad Appld Iformao Tchology h Spmbr 4. Vol. 67 No. 5-4 JATIT & LLS. All rghs rsrvd. ISSN: E-ISSN: b b + b b ad () b + b b, r, +,. () r r r I ohrwords, r sasfy h dffrc quao () oghr wh h al codos (). Thus, r ca b obad by solvg () wh h hlp of (). W ca pu h rsul h form of a horm as follows: Thorm 3..: If h roos of h quao ( z) z + b z +... b f + all l wh h u crcl, h h auorgrssv procss X +b X - +b X - + +b X -, ca b rprsd as a f movg avrag X br r., whr b r ar h roos of h dffrc quao b, r, +,. subc r + b r r o h al codos b, b + b,, b + bb b b. r 4. ILLUSTRATIONS W gv h followg llusraos o auhca our clam. Illusrao 4.: Cosdr h dffrc quao (3) 3 3 Hr p, <q <, σ 3, f (, l ), 3 3 l 3. Hr quao (3) ca b wr as b b b b + + b + (4) + b c wh h codos b b3 b c, b b4,ad b Ths s h rprsao of h dffrc quao auorgrssv procss. Codos (C), (C5) of horm 3.. ar sasfd. Hc all soluos of quao (3) ar ooscllaory. I fac,{ } s o such soluo of quao (3). Illusrao 4.: Cosdr h dffrc quao (5) ( ) ( 6 5 ) + Hr p, <q 5 3 <, τ, σ, f (, ) ( ) ( 6 5 ),.5 < <.9, l. Codo (C3) of horm 3.. s sasfd.hc all soluos of quao (5) ar ooscllaory. I fac,{ } s o such soluo of quao (5). Illusrao 4.3: Cosdr h dffrc quao ( ) + +. (6) Hr p, q >, σ, f (, l ), l. Codo (C) of horm 3.. s sasfd. Hc all soluos of quao (6) ar ooscllaory. I fac,{ } { } s o such soluo of quao (6). Illusrao 4.4: L {X()} b a MA procss of ordr gv by X r a +a - + +a -, a, whr { } s a purly radom procss. If h roos of h characrsc quao z +a z - + +a all l wh h u crcl, h X ca b rprsd as a auorgrssv procss of f ordr r r c X,whr h coffcs c r sasfy h r dffrc quao c a c a c, r + r r 8

6 Joural of Thorcal ad Appld Iformao Tchology h Spmbr 4. Vol. 67 No. 5-4 JATIT & LLS. All rghs rsrvd. ISSN: E-ISSN: r, +,.ad h al codos c. c + a, c + a c + a,.., c + ac a. 5. CONCLUTION AND FUTURE WORK I hs papr, w hav prsd cssary codos o prdc h asympoc bhavor of h m srs usg scod ordr dffrc of h combaos of obsrvaos obad from a gral m srs. Spcfc llusraos ar gv o auhca our clam. I fuur, w wll cosdr h gralzao hs modl o hghr ordr dffrc quao o sudy m srs aalyss. REFRENCES: []. R.P.Agarwal, Dffrc Equaos ad Iquals, Scod do, Marcl Dkkr,Nw York,. []. R.P.Agarwal, Mar Bohr, Sad R. Grac, DoalO Rga, Dscr Oscllao Thory, Hdaw, Nw York, 5. [3]. T.W.Adrso, Tm srs aalyss ad Forcasg: Th Bo-Jks Approach, Burworhs, Lodo, 976. [4]. Walr G.Klly ad Alla C.Prso, Dffrc Equaos - A Iroduco whapplcaos, d do, Acadmc Prss, Sa Dgo,. [5]. M.G.Kdall, Tm srs, C.Grff, Lodo, 96. [6]. J.W. Luo ad D.D. Baov, Oscllaory ad asympoc bhavor of scod-ordr uraldffrc quaos wh mama, Joural of Compuaoal ad Appld Mahmacs, Vol. 3, No.,, pp [7]. J.Mdh, Sochasc procsss, Nw Ag Iraoal Publshrs, Scod do, 6. [8]. J.Mdh, A o o h proprs of a s procdur for dscrmao bw wo yps of spcra of a saoary procss, Skad. Ac.,, 959, pp [9]. J.Mdh ad T.SubbaRao, Squal ad o-squal s procdurs for dscrmao bw dscr, couous ad md spcra of a saoary m srs, J. Id. Sa. Ass., 5, 967, pp. 8. []. J.Nyma, Idrmsm Scc ad w dmads for sascas, J. Am. Sa. Ass., 55, 96, pp []. EwaSchmdl, A applcao of masurs of ocompacss h vsgao of bouddss of soluos of scod-ordr ural dffrc quaos,aprl 4, 3, Vol. 9, No..hp:// om/co/3//9. []. B.Szmada, Proprs of soluos of hghr ordr dffrc quaos, Mahmacal ad Compur Modllg, Vol. 8, No., 998, pp. 95. [3]. E.Thadapa ad B.Slvara, Esc ad Asympoc Bhavor of No Oscllaory Soluos of Cra Nolar Dffrc Equaos, Far Eas Joural of Mahmacal Sccs (FJMS), 4 () 4, pp [4]. E.Thadapa ad P.Sudaram, O h asympoc ad oscllaory bhavor of soluos of scod ordr olar ural dffrc quaos, Ida Joural of pur ad appld Mahmacs, Vol. 6, No., 995, pp [5]. M-RogXu, Bao Sh ad Xao-Yu Zg, Asympoc bhavor for o-oscllaory soluos of dffrc quaos wh svral dlays h ural rm, Joural of Appld Mahmacs ad Compug, Vol. 7, No., 8, pp [6]. Zhguo Zhag, Jafg Ch ad Cashu Zhag, Oscllao of soluos for scodordr olar dffrc quaos wh olar ural rm, Compurs ad Mahmacs wh Applcaos, Vol. 4, No.,, pp [7]. Yu-Pg Zhao ad X-La Lu, Asympoc bhavor for ooscllaory soluos of olar dlay dffrc quaos, Iraoal Joural of Dffrc Equaos, Vol. 5, No.,, pp