The Asymptotical Behavior of Probability Measures for the Fluctuations of Stochastic Models

Transcription

1 The Asympoial Behavior of Probabiliy Measures for he Fluuaions of Sohasi Models JUN WANG CUINING WEI Deparmen of Mahemais College of Siene Beijing Jiaoong Universiy Beijing Jiaoong Universiy Beijing 44 PRChina CHINA Absra: - We onsider he fluuaions of shapes of wo phases boundaries of he one-dimensional saisial mehanis models By applying he heory of one-dimensional random walk he models of he wo phases boundaries are onsrued by assuming ha here is a speified value of he large area in he inermediae region of he wo phases boundaries Then we invesigae he asympoial behavior of he orresponding seuene of probabiliy measures desribing he saisial properies of he wo phases boundaries We show ha he limiing probabiliy measures oinide wih some ondiional probabiliy disribuion of erain Gaussian disribuion Furher we disuss he properies of fluuaions of phase separaion lines for he Ising model and we obain he asympoi properies of he wo inerfaes SOS model Key-Words: - Sohasi models; random phase boundaries; enral limi heory; random walk; Gibbs measure; Hamilonian Inroduion The problem of desripion of shapes of phase boundaries is a well-known problem in saisial mehanis sysems In reen years some researh work has been done o invesigae he saisial properies of he random phase boundaries for some saisial physis models for eample see Refs [- 8] In his paper we onsider he saisial limiing properies of he wo random phases boundaries model This work originaes in an aemp o desribe he fluuaions of he phase boundaries in wo random phases boundaries models (eg onedimensional wo random phases boundaries SOS model In Ref [] he saisial properies of random walks and he inerfae of Widom- Rowlinson model (ondiioned by fiing a large area under heir pahs and ondiioned by fiing he erminaing poin are onsidered and he enral limi heorem for hese ondiional disribuions is proved In [7] he inerfaes of superriial Ising model (see [9-] on he laie fraal---he Sierpinski arpe is sudied The similar problems arise in desribing he fluuaions of wo random phases boundaries models In he firs par of he presen paper wih he ondiions fied area of he inermediae layer and fied end poins in a wo random pahs model we sudy he limiing properies of he wo random phases boundaries see [] In he seond par of his paper he researh resuls of he firs par will be eended and improved he saisial properies of he inerfaes of SOS model and he wo-dimensional sohasi Ising model are sudied We show ha he heighs of he fluuaions of phase separaion lines of he Ising / / model our on a sale l (ln l for a large parameer and a large l ( he Ising model is onsidered on a reangle of horizonal side lengh l Then we disuss he asympoi properies of he wo inerfaes SOS model and obain he orresponding limiing resuls for he wo inerfaes SOS model In his paper we onsider he phase boundaries (or inerfaes models onsising of he inerfaes wihou overhangs and herefore is onfiguraions of he horizonal lengh are represened by se of heighs h Ζ { + + } Ζ A eah sie of he one dimensional laie Ζ we aah he variable of heighs h Ζ herefore he onfiguraions of he random inerfaes model on a horizonal se (wih he lengh of are represened by ses of heighs Ω { } { } h h for he simpliiy we assume The energy of ISSN: Issue 5 Volume 7 May 8

2 he onfiguraion { h} { h} he Hamilonian is deermined by ( + H ( h U h h i i U ( is a real-valued funion There are many possible hoies for he funion U ( his means ha he resuls of he presen paper an be eended o some oher inerfaes models For he sake of simpliiy we resri ourselves o he ase of ineger-valued heighs h Ζ e a posiive parameer be an inverse emperaure and he finie pariion funion of his sysem be h Z h Z [ ] Z ep H( h Then he orresponding Gibbs probabiliy disribuion on Ω is given by [ ] P h Z ep H ( h Ne we onsider he wo phases boundaries saisial mehanis model A eah sie of he one dimensional laie Ζ we aah wo variables of heighs h h Ζ herefore he onfiguraions of he random pahs model on a horizonal se are represened by ses of heighs h h h h for he simpliiy we also { } { } assume Now we define he inerfaes of he wo random inerfaes model as followings for [ ] h j hj j h ( h j h j+ j+ + j j j+ h j and h ( are defined similarly as above definiions For and le h h h h so we have h h and le e { } { } hen he Hamilonian of he model on he horizonal se of is given by H ( U ( + U ( i i U ( U ( are real-valued funions The pariion funion of he dynami sysem is defined as following Z ep [ H( ] is a posiive parameer alled an inverse emperaure The orresponding Gibbs probabiliy disribuion on is given by P Z ep H ( [ ] Thus we have he orresponding inerfaes j j ( ( and From above definiions { } { } an be seen as he seuenes of iid random variables respeively So he wo random inerfaes model has wo independen random SOS pahs ha is he model orresponds o he ensemble of wo independen self-avoiding pahs in [ ] Ζ saring from ( and ending a sies z in he line { } ( z ( y whih do no go bak in he horizonal direion Ne we inrodue he generaing funion of he heigh of he endpoins for one sep ha is for a fied le Q( ν + ν ( + e ep H ( Q( ν is independen of and < <+ Due o he independene of he random variables { } and { } hus Q ν ep + ν ep H and For ( ν R R we define Z ϕ( ν lim ln ep[ + ν ]ep[ H( ] Z by he Refs [][3][6] i is known ha his limi eiss if ( ν is in some neighborhood of he origin ISSN: Issue 5 Volume 7 May 8

3 The aim of his paper is o sudy he asympoes of fluuaions of he wo random inerfaes ondiioned by fiing a large area beween he wo random inerfaes Denoe by a a represening j he areas under he pahs j respeively and denoe by a a a represening he area of he inermediae layer beween he wo random inerfaes For a real and s assume ha d F s ϕ s s ( ( d F ( s ds a ( a > is some onsan Above ( is an imporan ondiion for his paper we will use his ondiion o fulfill our proof in he followings Then we sae he main resuls of his paper Theorem Assume ha for some δ( > and a > here eiss a real saisfying above < δ hen he proess ondiion ( and Y( ( ( F( s ds under P ( a a onverges weakly o he proess Y( ϕ" ( s ( s db s ondiioned ha Y( d ( { B( s } s he one dimensional sandard Brownian moion and a is he ineger par of a Remark In Theorem he model is only ondiioned by fiing a large area beween he wo random inerfaes and having he same saring endpoins The resuls an also be proved similarly for he wo random inerfaes wih fied value of area and he wo same endpoins Theorem e ϕ '( ν ϕ( ν and F ( s ϕ' ( ( s ( s Wih he same ondiions of Theorem he probabiliy disribuion of he random proess under P ( a a is onverges weakly o he orresponding probabiliy disribuion onenraed on he funion Y( F ( s ds Convergene of Probabiliy Measures for he Two Random Inerfaes Model In his seion we begin disussing he area beween he wo random pahs Then we show he some resuls abou he weak onvergene (see [] of random veor of he wo random inerfaes for he model Now we define he areas of a a a as followings ( ( a h a h a a a By he independene of { } { } and he generaion funion of he area a is defined by Qa { a } { H( } Z ep ep { ( ( ( } ep + Z ( ( ( Q e be a naural number and le { i } i be any se of real numbers suh ha < < < Se a random veor as ˆ a h h h h ( + Then for R we have ˆ ( ( H ( e e Z ( ( ; ( ; Q ISSN: Issue 5 Volume 7 May 8

4 ( ; ( [ ]( + i i For he real defined in ( and some small + onsan α > le R saisfy he following ondiions D { i i } : α < < + α < α α Ne we inrodue he orresponding uadrai form a ( + ( + mari denoe by V ( ˆ ( ( H ( Hess ln e e Z V ( is analyi in D α D α and aording o he definiion of Assume ha hen uniformly in and ha y we have ( ( V y y y R + suh yv y yv y as V( Hess ln Q ( ( s ( s ds and ( s ( s [ ]( s i i for s ˆ ( e P be he probabiliy disribuion of ˆ ( ( under P and Pˆ be given by ˆ z ˆ ep ˆ P z e P z E ( { } ( D α and z Z ( Z Z for all Denoe by ˆ E ( he orresponding epeaion funion for ˆ ( By he uniform boundedness of he E family of analyial funions V ( for all and D α all in aording o emma 6 and Proposiion 7 in Ref [] we have he following emma and emma emma e D α Then he random veor ˆ Y and ( ( ˆ ˆ ˆ E ( as onverges weakly o a Gaussian random veor ˆ Y of whih ovariane mari is given by V ( e g be he densiy funion of he Gaussian ( veor ˆ Y ( given in emma hen we have he following emma emma e z ( Z and ( hen for eah Z Z Z D α define ( ˆ( ˆ ( z Z y z E ( Then we have + 3/ˆ P ( z g y as uniformly in z ( Z and D α 3 Convergene of Finie Dimensional Disribuions In his seion we disuss he limiing properies of he random veor ˆ defined in Seion ( and show he proofs of Theorem Then we give he proof of Theorem in fa by using he proofing mehod of Theorem we an prove Theorem Proof of Theorem In Seion he random ( veor ˆ ( is given Firs we onsider he onvergene of he finie-dimensional disribuion of he random veor ˆ Y defined in ( emma e be a speial seuene in D α suh ha ( ( is defined in ( and saisfies he following ondiion d ln Q ε( a α d ( by (( i an be proved ha as here we omi he proof e ˆ ( ( H( e ϕ( ; ln e i Z and denoe by ISSN: Issue 5 Volume 7 May 8

5 ( ( ; lim ( ; ϕ ϕ for By he uniform boundedness of D α Hess ϕ we have E ˆ ˆ ( ( ( ˆ( ( ˆ a E h h E ( h h ( ϕ( ; ( ( ϕ ( ; ο( + By emma we have for < a < b < j j j lim Pˆ y a b j z a ( ( j j j ˆ ( P ( y j aj b j j z a g ( y y [ ] dy y a b a b g ( y y dy y lim R Aording o emma le ˆ Y Y Y Y be a Gaussian random veor wih disribuion densiy g ( y y Then is ovariane mari is given by E Y( j Y( K j k ϕ " ( ( s ( s ds j E YY " ( j ϕ s s ds " E Y ϕ ( ( s ( s ds a b min { a b} { { } [ ] } for jk means ha Y Y( This is a Gaussian random proess wih ovariane mari given above for every In above proof we suppose ha i i h h i i for Similarly o Ref [] he above argumen is also rue if we replae i i wih ( i ( i for every i Then he disribuion of ˆ ( ( ε under P ( α a onverges weakly o he orresponding disribuion ( of Gaussian random veor ˆ Y ( Seondly he ighness of above ondiional disribuion of he random proess Y ( should be disussed see [] Following he similar argumen of Seion 3 in Ref [] we an prove a suffiien ondiion for he ighness of he onsidered proess Y ( So by he heory of weak onvergene (see [] ogeher wih he firs par of his proof his omplees he proof of Theorem Remark Aording o he argumens of [] and wih he resuls of Theorem he probabiliy disribuion of he random proess ( under P ( α a onverges weakly o he orresponding disribuion onenraed on he funion F ( s ds Proof of Theorem e be a naural number and le { i i } be any se of real numbers suh ha < < < Se a random veor as e ( ˆ ( a h h be a speial seuene in suh ha D α ( ( is defined in ( and is defined in ( Then we have he orresponding funion as following ϕ ; ( ˆ ( ln H ( e i e Z ln ( ( ; Q ( ; ( ; ( [ ]( i ( ; ( + i + For any R saisfy he following ondiions D { : α < < + α i < α α } e ϕ ; lim ϕ ; ( ( ISSN: Issue 5 Volume 7 May 8

6 for and Eˆ ( is he orresponding D α epeaion funion for ˆ ( uniform boundedness of Hess ϕ we have ( ˆ ˆ E ( ˆ ˆ a E h E h ( ϕ( ; ϕ ; + ο j and Eˆ ( h ( ( ( ( ln Q ; ; i For he random veor ˆ ( By he by using he mehods of emma 6 and Proposiion 7 in [] we an have he similar resuls as ha of emma and emma Then following he seps in he proof of Theorem we an prove ha he probabiliy disribuion of he random proess ( F ( s ds under P ( α a onverges weakly o some Gaussian disribuion Thus by Remark he probabiliy disribuion of he random proess ( under P ( a a onverges weakly o he orresponding probabiliy disribuion onenraed on he funion Y( F ( s ds This omplees he proof of Theorem Aording o he resuls of Theorem and Theorem we have he following Corollary Corollary Suppose ha he definiions and ondiions of Theorem hold hen he probabiliy disribuion of he random proess under P ( α a onverges weakly o he orresponding probabiliy disribuion onenraed on he funion F ( d F ( d Proof The random proess ( α a an be wrien as ( α a ( ( ( α a + ( ( α a For he firs erm of above euaion under P ( α a and by Theorem and Remark we have ha he probabiliy disribuion of he random proess ( ( ( α a onverges weakly o he orresponding probabiliy disribuion of he funion F ( d For he seond erm of above euaion under P ( α a and aording o Theorem and Remark he probabiliy disribuion of he random proess ( α a onverges weakly o he orresponding probabiliy disribuion of he funion F ( d This omplees he proof of Corollary 4 The Fluuaions of SOS Model and Ising Model In his seion we disuss he relaions beween he wo random inerfaes model and he wo inerfaes SOS model and Ising model The saisial properies of he inerfaes of SOS model and Ising model are sudied in his seion The S Hamilonian ( model has he same definiion of ( H h h of wo inerfaes SOS H h h in Seion Bu he pariion funion of wo inerfaes SOS model is given by S S Z ep H ( h h h h and aording o he definiions in Seion we have he orresponding pariion funion S S Z ep H ( denoe ha for all From above definiions for he wo inerfaes SOS model he wo inerfaes of he model don' inerse so ha he wo inerfaes are no ISSN: Issue 5 Volume 7 May 8

7 independen The orresponding probabiliy measure is defined as following S S P Z ep H( (3 on he se {( : } In his paper we disuss he mos popular ferromagnei model ha is he sohasi Ising model Firs we disuss he fluuaions of wodimensional Ising model Sine he SOS model is he simple ase of Ising model so our resuls of Ising model in his paper an be done similar o he SOS model e Z be he usual wodimensional suare laie wih sies u ( u u euipped wih he l -norm: u u + u Given Z Ω { + } is he onfiguraion spae An elemen of Ω { + } will usually denoe by { ( u: u } Whenever onfusion does no arise we will also omi he subsrip in he noaion Given a boundary ondiion we onsider he Hamilonian H ( ( ( u ( u' ( uu ' u u' ( ( u ( u' ( uu ' u u' The Gibbs measure assoiaed wih he Hamilonian is defined as ep H ep H ( Ω is a parameer Noe ha we use ( (no P o denoe he probabiliy measure for he Ising model The sohasi dynamis whih is sudied in he presen paper is defined by he Markov generaor ( u A f ( u f( f( u aing on ( d u Ω + ( v if u v u and ( v if v u u ( is he ransiion raes for he proess ([9-] saisfying neares neighbour ineraions araiviy boundedness and deailed balane ondiion u ( u u ( u ( ( e Z be a reangle of side lengh l (horizonal size and m For he wo-dimensional Ising model (see [9-] by using he ehniues of orrelaion funions for esimaing he fluuaion of phase separaion (or inerfae line when I sin g I sin g > ( is he riial poin of Ising model we an prove ha wih probabiliy larger han ep ( lnl he inerfae has a heigh less han ( ( lln l / l large enough and ( ( are posiive onsans e ΩQ lm be he onfiguraion spae of he Ising model and Q lm be he orresponding Gibbs measure wih he boundary ondiion is defined by if u m + + if u m for u ( u u Z e Z be he dual laie of Z ie Z Z + (/ / For uv R le [ uv ] be he losed segmen wih uv as is endpoins The edges of Z ( Z are hose e [ u v] wih uv neares neighbours in Z ( Z Given an edge e of Z e is he uniue edge in Z ha inerses e We denoe by B he se of edges suh ha boh endpoins are in and by B he se of all edges wih a leas one endpoin in Given Z we le Z \ and define as he se of all u Z suh ha du ( du ( inf{ u v : v } The se of he dual edges is defined as B { e : e B} The inerior and eerior boundaries of are defined by in { u : v u v } e { u : v u v } and in e are defined in he similar way For simpliiy we all an edge in Z by a bond so ha we an disinguish i from edges in Z We say ha a neighbouring pair u and v in Z are separaed by a bond e if he edge e [ u v] inerses e e Z and { + } Z be fied for every onfiguraion Ω we denoe by Γ ( he olleion of all bonds separaing neighbouring sies u and v suh ha: (i uv and ( u ( v or (ii u v and ( u ( v e ISSN: Issue 5 Volume 7 May 8

8 We divide Γ ( ino onneed omponens Furher we use he onvenion ha any pair of orhogonal bonds ha inerse in a given sie u of he dual laie Z are a linked pair of bonds iff hey are boh on he same side of he fory-five degrees line aross u hen we regard ha wo linked pairs a u are no onneed a u By his onvenion eah onneed omponen of Γ ( say Γ has he following properies: (i if u \ in hen he number of bonds in Γ ha inerse u is always even; (ii bonds in Γ an be ordered as e e en so ha e i and e i+ have a ommon vere for every i and if Γ has a poin u a whih 4 bonds in Γ whih inerse u hen here are i j suh ha hese 4 bonds are divided ino wo linked pairs { ei ei+ } and { ej ej+ } We all hese omponens of Γ ( by onours in (wih boundary ondiion If for any u Z he number of bonds in he onour Γ whih inerse u is even hen we all Γ a losed onour A onour whih is no losed is alled by an open onour The lengh Γ of a onour is simply he number of bonds in Γ Now we give he following emma 3 and emma 4 They are imporan for us o esimae he heighs of he inerfaes emma 3 For he wo-dimensional sohasi Ising model le Q lm be defined as above and I sin g le > For some k ( > se / [ ( ln / / m k l l ] k [ k( l ( ln l / /] Suppose ha Q k {( u u : u m 3 k} hen here are ( > and l l( > independen of Q lm suh ha for all l > l and u Q k we have Q lm (( Q ep lm F ln l F Q lm is he even 3m ΩQ : Γ { : } lm open u Ql m u 6 denoe hose open onours produed F and Γ open by he onfiguraion ondiion on Q lm Ω Q lm wih boundary Proof The proof of emma 3 depends on he esimaes of he heighs of he inerfaes for he Ising model By he emma 6 of [8] for I sin g > and some large onsan M > when l is large enough we have Q Γopen( S( A B: M ln l lm κ( M l lm and ( M ep ln (4 A ( l m B ( κ > is a posiive parameer le S( A B: M ln l { u : u A + u B A B + M lnl} Aording o he definiion of Q lm and Q k by he ompuaion of S( A B: M ln l and above (4 he fluuaions of phase separaion line our on a sale / / l (ln l ha is here are k > (dependen on M suh ha Γ { u Q : u 3 m/6} ( ep ( ln l open l m > This ineualiy proves he ineualiy of emma 3 emma 4 For he wo-dimensional sohasi Ising model le Q lm be defined as above and I sin g le > For some k ( > se / [ ( ln / / m k l l ] k [ k( l ( ln l / /] Suppose ha Q k {( u u : u m 3 k} hen here are ( > and l l( > independen of Q lm suh ha for all l > l and u Q k we have ( ( u ( ( u Q ( F lm + F Q lm is he even F 3m Q lm Ω : { Q Γ : } lm open u Ql m u 6 and Γopen ( denoe hose open onours produed by he onfiguraion Ω Q lm wih boundary ondiion on Q lm Furher we have + ( u ( u Q ( F lm ep ( + Q lm ln l is he Gibbs measure wih he plus boundary ondiion on Q lm ISSN: Issue 5 Volume 7 May 8

9 Proof e an wrie F Q lm be he even defined as above We Q ( ( u lm Q ( u F lm Q lm Q F lm ( F Q lm FKG ineualiy + Q ( ( u ( F lm is jus he omplemen even By he ( ( u F ( F + ( ( u Then we have he differene ( u ( ( u + Q ( F lm Combing he resul of emma 3 his ineualiy proves he ineualiy of emma 4 Remark 3 emma 3 and emma 4 are proved for he wo-dimensional sohasi Ising model hey desribe he saisial properies of he inerfaes of he Ising model The simple ase of his problem arises in he one-dimensional SOS model Through he similar argumens in he proof of emma 3 and emma 4 we an have he similar resul as ha of emma 3 and emma 4 for onedimensional SOS model ha is he inerfaes of SOS model have a heigh less han ( ( lln l / wih large probabiliy In above Remark 3 we disuss he inerfae heigh for one-inerfae SOS model The aim of his paper is o sudy wo random pahs model and wo inerfaes SOS model From Seion o Seion 3 we have sudied he inerfae of he wo random pahs model ondiioned on a fied area in he inermediae layer and fied end poins In his Seion by using emma 3 emma 4 and Remark 3 we sudy he relaions beween he wo random pahs model and he wo inerfaes SOS model In he definiions of Seion wih he saring poins and we disussed he wo random pahs model wih he pariion funion of Z ep H ( While in his Seion we modify he end poins of he model e Ψ denoe he even M M ln > is a large posiive onsan ( 4( j The random pahs j are defined in Seion and le ϒ denoe he even ha he j random pahs and j don' inerse eah oher on hen we have he following emma 5 emma 5 For he wo random pahs model defined in Seion here are ( > ( > and > suh ha for all > and for all > ( ep ( P ϒ Ψ ln P is he orresponding probabiliy measure for wo inerfaes SOS model whih is defined in (3 Proof The proof of emma 5 follows direly from emma 3 emma 4 Remark 3 and he ondiion M( > 4( This lemma shows ha wih large probabiliy he wo random pahs don' inerse eah oher e ϒ Ψ ( ( ( ( ( ( and P P ( ϒ Ψ be he orresponding ondiional probabiliy disribuion of he random proess ( ( ( Aording o above preparaion and emma 5 we have he following Corollary Corollary Wih he same ondiions of emma 5 we have he following lim P G Ψ { (( ( ( ( ( ( G [ a a ] [ b b ] a i } P G < < b < for i i From he definiion of he proess i is known ha he proess ( ( ( ( ( is a ondiional wo inerfaes SOS model (wih he speial fied end poins Corollary shows a limiing relaion beween he wo random pahs model and he ondiional wo inerfaes SOS model This resul is useful o sudy he asympoi properies of he wo inerfaes SOS model by using he resuls of wo random ISSN: Issue 5 Volume 7 May 8

10 pahs model for eample we onsider he wo inerfaes SOS model wih a large fied area beween he wo inerfaes e 5 Conlusion In his paper we sudied he saisial properies of he wo random inerfaes model Under some ondiions ha here is a speified value of he large area in he inermediae region of he wo random inerfaes Theorem shows he weak onvergene of he fluuaions for he wo random inerfaes In Seion 4 he researh resuls in Seion -3 are eended and improved for he wo inerfaes SOS model The resuls of he presen paper an also be applied o oher fields for eample see [3-5] Aknowledgemens The auhors are suppored in par by Naional Naural Siene Foundaion of China Gran No7776 BJTU Foundaion No6M44 The auhors would like o hank ZQ Zhang and BT Wang for heir kind ooperaion on his researh work Referenes: [] Y Higuhi J Murai and J Wang The Dobrushin-Hryniv Theory for he Two-Dimensional aie Widom-Rowlinson Model Advaned Sudies in Pure Mahemais vol 39 pp [] Y Higuhi On some imi Theorems Relaed o he Phase Separaion ine in he Two Dimensional Ising Model Z Wahrsheinlihkeisheorie verw Gebiee vol 5 pp [3] R Dobrushin R Koeky and S Shlosman Wullf Consruion A Global Shape from oal Ineraion Providene Rhode Island: Amerian Mahemaial Soiey 99 [4] J Wang and S Deng Fluuaions of inerfae saisial physis models applied o a sok marke model Nonlinear Analysis: Real World Appliaion vol 9 pp [5] J Wang The Speral Gap of Two Dimensional Ising Model wih a Hole: Shrinking Effe of Conours J Mah Kyoo Univ (JMKYAZ vol 39 no 3 pp [6] J Wang The saisial properies of he inerfaes for he laie Widom-Rowlinson model Applied Mahemais eers vol 9 pp [7] J Wang Superriial Ising Model on he aie Fraal---he Sierpinski Carpe Modern Physis eers B vol pp [8] C E Pfiser and Y Velenik Inerfae surfae ension and reenran pinning ransiion in he D Ising model Commun Mah Phys vol 4 pp [9] M F Chen From Markov Chains To Non- Euilibrium Parile Sysems World Sienifi 99 [] T M igge Ineraing Parile Sysems Berlin: Springer-Verlag 985 [] R S Ellis Enropy arge Deviaions and Saisial Mehanis New York: Springer-Verlag 985 [] P Billingsley Convergene of probabiliy measures New York: John Wiley & Sons 968 [3] Q D i and J Wang Saisial Properies of Waiing Times and Reurns in Chinese Sok Markes WSEAS Transaions on Business and Eonomis vol3 pp [4] M F Ji and J Wang Daa Analysis and Saisial Properies of Shenzhen and Shanghai and Indies WSEAS Transaions on Business and Eonomis vol 4 pp [5] J Wang and Q Y Wang The Saisial Properies of Fluuaions of Inerfaes for Voer Model Inernaional Journal of Mahemais and Compuers in Simulaion vol pp ISSN: Issue 5 Volume 7 May 8