Kefàlaio 1

Eisagwg† stic StoqastikËcDiadikas–ec

1.1 Eisagwg†

Sto eisagwgikÏ màjhma Pijanot†twn maja–nei kane–c ta basikà antike–mena thc jewr–ac twn Pijano-t†twn, touc q∏rouc pijanÏthtac (probability spaces) kai tic tuqa–ec metablhtËc (random variables).Me ta antike–mena autà mpore– na melethje– h katàstash enÏc sust†matoc anàloga me to apotËle-sma enÏc peiràmatoc t‘qhc kai gi' autÏn to lÏgo h Ënnoia tou qrÏnou sun†jwc den upeisËrqetai staprobl†mata pou anal‘ontai. SkopÏc aut∏n twn shmei∏sewn e–nai na eisagàgoun ton anagn∏sth sthmontelopo–hsh kai sthn anàlush fainomËnwn ta opo–a exel–ssontai ston qrÏno me trÏpo pou emfan–zeikàpoia tuqaiÏthta. Ta majhmatikà antike–mena pou montelopoio‘n tËtoia fainÏmena e–nai oi stoqastikËcdiadikas–ec (stochastic processes), me thn perigraf† kai thn kataskeu† twn opo–wn ja asqolhjo‘me stopr∏to kefàlaio.

1.2 KatanomËc peperasmËnhc diàstashc

Upàrqoun diàforoi isod‘namoi trÏpoi na or–sei kane–c mia stoqastik† diadikas–a. Se kàje per–ptwshqreiazÏmaste Ëna s‘nolo X, pou onomàzetai q∏roc katastàsewn (state space) kai sto opo–o an†kounoi timËc pou pa–rnei h stoqastik† diadikas–a kai Ëna s‘nolo T pou e–nai sun†jwc -qwr–c autÏ na e–naikanÏnac- Ëna s‘nolo qrÏnwn, p.q.

T = [0,+1) † T = N0 = {0, 1, 2, . . .}.

O aplo‘steroc trÏpoc na or–sei kane–c mia stoqastik† diadikas–a, orismËnh sto T kai me timËc sto Xe–nai wc mia sullog† apÏ tuqa–ec metablhtËc {Xt}t2T , orismËnec se Ënan q∏ro pijanÏthtac (⌦,F ,P),me timËc sto X. Se aut† thn per–ptwsh, gia kàje t 2 T , h tuqa–a metablht† Xt : ⌦ 3 ! 7! Xt(!) 2 Xperigràfei thn katàstash tou sust†matoc pou mac endiafËrei thn qronik† stigm† t kai h sullog† toucperigràfei thn katàstash tou sust†matoc kàje qronik† stigm† sto s‘nolo T .

'Enac àlloc trÏpoc na dei kane–c mia stoqastik† diadikas–a e–nai na jewr†sei gia kàje ! 2 ⌦ thn troqiàthc stoqastik†c diadikas–ac, kaj∏c o de–kthc t metabàlletai sto T , na dei dhlad† thn t 7! Xt(!) wc miasunàrthsh apÏ to T sto X. Se aut† thn per–ptwsh h stoqastik† diadikas–a e–nai mia tuqa–a sunàrthsh,dhlad† mia tuqa–a metablht† orismËnh sto ⌦ me timËc sto s‘nolo XT twn sunart†sewn apÏ to T sto X.

1

Paràdeigma 1 Str–boume Ëna kËrma treic forËc. Xekin∏ntac apÏ to mhdËn kànoume Ëna b†ma proc tadexià kàje forà pou fËrnoume kefal† kai Ëna b†ma proc ta aristerà kàje forà pou fËrnoume gràmmata.Mporo‘me na perigràyoume th jËsh mac wc mia stoqastik† diadikas–a orismËnh sto s‘nolo qrÏnwnT = {0, 1, 2, 3} me timËc ston q∏ro katastàsewn Z. Pràgmati, an gia k = 0, 1, 2, 3 sumbol–zoume meXk th jËsh mac metà apÏ k striy–mata, tÏte oi {Xk}k2T e–nai tuqa–ec metablhtËc me timËc ston Z, poumporo‘me na tic or–soume ston deigmatikÏ q∏ro

⌦ = {KKK,KK�,K�K,K��,�KK,�K�,��K,���}.

'Eqoume loipÏn X0(!) = 0 gia kàje ! 2 ⌦ kai

X1(!) =

(+1, an ! 2 {KKK,KK�,K�K,K��}�1, an ! 2 {�KK,�K�,��K,���},

X2(!) =

8><

>:

+2, an ! 2 {KKK,KK�}0, an ! 2 {K�K,K��,�KK,�K�}�2, an ! 2 {��K,���},

X3(!) =

8>>>><

>>>>:

+3, an ! = KKK

+1, an ! 2 {KK�,K�K,�KK}�1, an ! 2 {��K,�K�,K��}�3, an ! = ���.

An do‘me thn –dia stoqastik† diadikas–a wc mia tuqa–a metablht† me timËc ston ZT Ëqoume Ïti

X·(KKK) = (0, 1, 2, 3), X·(KK�) = (0, 1, 2, 1), X·(K�K) = (0, 1, 0, 1) k.lp.

Parathr†ste Ïti kàje ! 2 ⌦ apeikon–zetai se mia troqià m†kouc |T | = 4 sto Z. 2

Sto prohgo‘meno paràdeigma kataskeuàsame mia stoqastik† diadikas–a perigràfontac ton deigmatikÏq∏ro ston opo–o thn Ëqoume or–sei kai d–nontac ton t‘po thc, ! 7! X·(!) 2 XT . Sthn jewr–a twnPijanot†twn Ïmwc ta erwt†mata pou mac endiafËroun den aforo‘n to po‘ kai p∏c Ëqoume or–sei miatuqa–a metablht†, allà tic statistikËc thc idiÏthtec, dhlad† to pÏso pijanÏ e–nai na breje– h tuqa–ametablht† se diàfora upos‘nola tou q∏rou entÏc tou opo–ou pa–rnei timËc. Aut† h plhrofor–a d–netaiapÏ thn katanom† thc tuqa–ac metablht†c. Gi' autÏ kai sun†jwc, Ïtan perigràfoume mia tuqa–a meta-blht†, den kànoume anaforà ston deigmatikÏ q∏ro ⌦ ston opo–o e–nai orismËnh kai ston t‘po thc, allàsthn katanom† thc. AutÏ ja prospaj†soume na kànoume t∏ra kai gia mia stoqastik† diadikas–a. Japrospaj†soume dhlad† na thn perigràyoume mËsw twn statistik∏n idiot†twn thc.

Ja †tan àrage arketÏ na perigràyoume thn katanom† thc tuqa–ac metablht†c Xt, gia kàje t 2 T ; ToepÏmeno paràdeigma mac didàskei pwc h apànthsh e–nai arnhtik†.

Paràdeigma 2 'Estw Ïti Ëqoume or–sei se kàpoion q∏ro pijanÏthtac mia akolouj–a {Zk}k2N0 apÏanexàrthtec, isÏnomec tuqa–ec metablhtËc me tupik† kanonik† katanom†. Or–zoume tic parakàtw d‘odiadikas–ec diakrito‘ qrÏnou {Xn}n2N0 kai {Yn}n2N0 .

Xn =pnZ0, gia n 2 N0 kai Y0 = 0, Yn =

nX

k=1

Zk, gia n 2 N.

EfÏson Z0 ⇠ N (0, 1), Ëqoume Xn =pnZ0 ⇠ N (0, n), gia kàje n 2 N. EpiplËon, efÏson oi {Zk}k2N

e–nai anexàrthtec tupikËc kanonikËc, tÏte kai h Yn ja akolouje– kanonik† katanom† gia kàje n 2 N, me

2

0 10 20 30 40 50

n

0 10 20 30 40 50

n

XY

Sq†ma 1.1: TupikËc troqiËc twn diadikasi∏n {Xn}n kai {Yn} tou Parade–gmatoc 2.

mËsh tim† E⇥Yn

⇤= nE

⇥Z1

⇤= 0 kai diasporà V (Yn) = nV (Z1) = n. 'Eqoume epomËnwc Ïti Yn ⇠ N (0, n)

gia kàje n 2 N. Parathr†ste Ïti kàje qronik† stigm† n 2 N0 oi tuqa–ec metablhtËc Xn kai Yn Ëqounthn –dia katanom†. ApÏ thn àllh pleurà, to Sq†ma 1.1 e–nai endeiktikÏ tou pÏso diafËroun oi troqiËc twnd‘o stoqastik∏n diadikasi∏n gia mia tupik† pragmatopo–hs† touc ! 2 ⌦. Parathr†ste akÏma Ïti, anxËrei kane–c thn tim† X1(!), mpore– na anakataskeuàsei Ïlh thn troqià X·(!). Ant–jeta, oi prosaux†seicthc {Yn}n metà th qronik† stigm† n = 1 e–nai anexàrthtec thc Y1. 2

E–dame loipÏn Ïti d‘o stoqastikËc diadikas–ec, pou kàje qronik† stigm† t 2 T Ëqoun thn –dia katanom†,mpore– na Ëqoun troqiËc me pol‘ diaforetikà poiotikà qarakthristikà. AutÏ sumba–nei giat–, perigràfo-ntac mÏno thn katanom† twn Xt, gia kàje t 2 T , den d–noume kam–a plhrofor–a gia th metax‘ toucexàrthsh. H plhrofor–a gia thn allhlexàrthsh tuqa–wn metablht∏n br–sketai sthn apÏ koino‘ kata-nom† touc (joint distribution). EpomËnwc, opoiad†pote pl†rhc perigraf† twn statistik∏n idiot†twnmiac stoqastik†c diadikas–ac ja prËpei na emperiËqei thn apÏ koino‘ katanom† twn tuqa–wn metablht∏nXt1 , Xt2 , . . . , Xtk , gia kàje k 2 N kai gia kàje diatetagmËnh k-àda qrÏnwn F = (t1, t2, . . . , tk) ⇢ T k.Kàje tËtoia katanom† e–nai Ëna mËtro pijanÏthtac µF ston Xk, pou or–zetai mËsw thc

µF

⇥A⇤= P

⇥(Xt1 , . . . , Xtn) 2 A

⇤(1.1)

gia katàllhla s‘nola A 2 Xk. Ja sumbol–zoume me S(T ) to s‘nolo twn peperasmËnwn diatetagmËnwnuposunÏlwn tou T , dhlad†

S(T ) =[

k2NT k =

[

k2N{(t1, . . . , tk) : ti 2 T, i = 1, . . . , k}.

Me àlla lÏgia to S(T ) apotele–tai apÏ ta peperasmËna diatetagmËna s‘nola oswnd†pote kai opoiwn-d†pote qrÏnwn.

3

OrismÏc: Onomàzoume th sullog† {µF }F2S(T ) Ïlwn twn katanom∏n thc (1.1) oikogËneia twn katano-m∏n peperasmËnhc diàstashc thc stoqastik†c diadikas–ac {Xt}t2T .

Mpore– na apodeiqje– Ïti h oikogËneia twn katanom∏n peperasmËnhc diàstashc miac stoqastik†c diadika-s–ac {Xt}t2T prosdior–zei pl†rwc tic statistikËc thc idiÏthtec. AutÏ e–nai d‘skolo na g–nei antilhptÏ memajhmatik† austhrÏthta, qwr–c thn orolog–a thc Jewr–ac MËtrou. Gia tic anàgkec auto‘ tou maj†ma-toc e–nai arketÏ na jumÏmaste Ïti mporo‘me na perigràyoume thn katanom† miac stoqastik†c diadikas–acprosdior–zontac Ïlec tic katanomËc peperasmËnhc diàstashc. An autËc e–nai gnwstËc, mporo‘me naapant†soume opoiod†pote pijanojewrhtikÏ er∏thma pou aforà th diadikas–a, qwr–c na qreiàzetai nagnwr–zoume se poion deigmatikÏ q∏ro e–nai orismËnh † poioc e–nai o t‘poc thc.

Paràdeigma 3 Ja qarakthr–zoume mia stoqastik† diadikas–a wc diadikas–a Gauss, an kàje katanom†peperasmËnhc diàstashc µF e–nai kanonik† se |F | diastàseic. EfÏson kàje poludiàstath kanonik†katanom† prosdior–zetai apÏ to diànusma twn mËswn tim∏n kai ton p–naka sundiakumànsewn, gia naperigràyoume mia diadikas–a Gauss {Xt}t2T , arke– na perigràyoume tic sunart†seic mËshc tim†c kaiautosusqËtishc

m : T ! R, me m(t) = E⇥Xt

⇤kai ⇢ : T ⇥ T ! R, me ⇢(s, t) = Cov(Xs, Xt).

H tupik† k–nhsh Brown e–nai mia diadikas–a Gauss, orismËnh sto T = [0,+1) me timËc sto R gia thnopo–a

m(t) = 0 kai ⇢(s, t) = s ^ t = min{s, t}, gia kàje s, t � 0.

To Ïti upàrqei tËtoia diadikas–a den e–nai kajÏlou autonÏhto. Mesolàbhsan e–kosi tr–a qrÏnia anàmesasthn pr∏th euretik† qr†sh aut†c thc diadikas–ac apÏ ton Bachelier kai sth majhmatikà austhr† kata-skeu† thc apÏ ton Wiener to 1923.

Gia kàje t � 0, h katanom† thc tuqa–ac metablht†c Xt e–nai kanonik†, afo‘ h Xt e–nai diadikas–a Gauss.EpiplËon, E

⇥Xt

⇤= m(t) = 0 kai V (Xt) = Cov(Xt, Xt) = ⇢(t, t) = t, àra Xt ⇠ N (0, t). EidikÏtera

X0 = 0.

Ac do‘me t∏ra poia katanom† akolouje– h prosa‘xhsh Xt�Xs thc k–nhshc Brown s' ena diàsthma (s, t].To ze‘goc (Xs, Xt) akolouje– kanonik† katanom† se d‘o diastàseic, afo‘ h {Xt}t�0 e–nai diadikas–aGauss. H Xt � Xs prok‘ptei apÏ tic Xs, Xt mËsw enÏc grammiko‘ metasqhmatismo‘, àra akolouje–kanonik† katanom†. E‘kola blËpei kane–c Ïti E

⇥Xt �Xs

⇤= m(t)�m(s) = 0, en∏

V (Xt�Xs) = Cov(Xt�Xs, Xt�Xs) = Cov(Xt, Xt)�2Cov(Xt, Xs)+Cov(Xs, Xs) = t�2s+s = t�s.

EpomËnwc Xt �Xs ⇠ N (0, t� s).

Jewr†ste t∏ra 0 r < s < t. Ja de–xoume Ïti oi prosauxhseic Xt �Xs kai Xs �Xr e–nai anexàrthtectuqa–ec metablhtËc. H apÏ koino‘ katanom† twn (Xr, Xs, Xt) e–nai tridiàstath kanonik† me diànusmamËswn tim∏n

�E⇥Xr

⇤,E

⇥Xs

⇤,E

⇥Xt

⇤�= (0, 0, 0) kai p–naka sundiakumànsewn

⌃(r, s, t) =

0

@r r rr s sr s t

1

A .

EfÏson oi (Xt � Xs, Xs � Xr) prok‘ptoun apÏ tic (Xr, Xs, Xt) mËsw enÏc grammiko‘ metasqhmati-smo‘, akoloujo‘n ki autËc kanonik† katanom† (se d‘o diastàseic). Arke– loipÏn na de–xoume Ïti e–naiasusqËtistec.

Cov(Xt�Xs, Xs�Xr) = Cov(Xt, Xs)�Cov(Xt, Xr)�Cov(Xs, Xs)+Cov(Xs, Xr) = s�r�s+r = 0.

Sto Sq†ma 1.2 fa–netai mia tupik† troqià thc k–nhshc Brown. 2

4

Sq†ma 1.2: Tupik† troqià thc k–nhshc Brown.

1.3 Diadikas–ec se diakritÏ qrÏno kai q∏ro

Sto megal‘tero mËroc aut∏n twn shmei∏sewn ja melet†soume stoqastikËc diadikas–ec se diakritÏ qrÏnome timËc se Ënan arijm†simo q∏ro katastàsewn X. Se aut† thn paràgrafo ja do‘me p∏c mporo‘me naperigràyoume e‘kola tic katanomËc peperasmËnhc diàstashc tËtoiwn stoqastik∏n diadikasi∏n.

'Eqontac epilËxei T = N0, prokeimËnou na perigràyoume Ïlec tic katanomËc peperasmËnhc diàstashc,arke– na perigràyoume thn apÏ koino‘ katanom† twn tuqa–wn metablht∏n (X0, X1, . . . , Xn) gia kàjen 2 N0. Pràgmati, gia kàje F = (n1, . . . , nk) 2 Nk

0, an jËsoume n = max{n1, . . . , nk}, tÏte mporo‘mena pàroume thn µF , dhlad† thn apÏ koino‘ katanom† twn (Xn1 , . . . , Xnk), wc mia katàllhlh perij∏riakatanom† thc apÏ koino‘ katanom†c twn (X0, X1, . . . , Xn).

EpiplËon, efÏson o X e–nai arijm†simoc, o Xk e–nai kai autÏc arijm†simoc gia kàje k 2 N kai epomËnwcÏlec oi katanomËc peperasmËnhc diàstashc e–nai diakritËc katanomËc. Mporo‘me loipÏn na tic peri-gràyoume mËsw thc sunàrthshc màzac pijanÏthtàc (s.m.p.) touc. SugkekrimËna, an Fn = (0, 1, . . . , n),h s.m.p. thc katanom†c µFn d–netai, gia kàje ⌘ = (x0, . . . , xn) 2 Xn+1, apÏ thn

Qn(⌘) = µFn

⇥{⌘}

⇤= P

⇥(X0, . . . , Xn) = ⌘

⇤= P

⇥X0 = x0, . . . , Xn = xn

⇤. (1.2)

EpomËnwc, sthn per–ptwsh pou Ëqoume mia stoqastik† diadikas–a se diakritÏ qrÏno T = N0 me timËc s'Ënan arijm†simo q∏ro katastàsewn X, prokeimËnou na perigràyoume Ïlec tic katanomËc peperasmËnhcdiàstashc thc diadikas–ac arke– na perigràyoume thn Qn(⌘) thc sqËshc (1.2), gia kàje n 2 N0 kai kàje⌘ 2 Xn+1. S' autÏ to shme–o ac jumhjo‘me ton pollaplasiastikÏ t‘po gia ta mËtra pijanÏthtac. AnA0, A1, . . . , Ak e–nai endeqÏmena enÏc deigmatiko‘ q∏rou, tÏte

P⇥A0 \ · · · \An

⇤= P

⇥A0

⇤P⇥A1

��A0⇤P⇥A2

��A0 \A1⇤· · ·P

⇥An

��A0 \ · · · \An�1⇤. (1.3)

5

An sthn parapànw sqËsh epilËxoume Aj = {Xj = xj}, j = 0, 1, . . . , n, Ëqoume Ïti

Qn(⌘) = P⇥X0 = x0

⇤P⇥X1 = x1

��X0 = x0⇤· · ·P

⇥Xn = xn

��X0 = x0, . . . , Xn�1 = xn�1⇤. (1.4)

O pr∏toc Ïroc tou dexio‘ mËlouc mpore– na upologiste– apÏ thn katanom† thc tuqa–ac metablht†c X0.O genikÏc Ïroc tou dexio‘ mËloc e–nai thc morf†c

P⇥Xk = xk

��X0 = x0, . . . , Xk�1 = xk�1

⇤

kai mpore– na upologiste– apÏ th desmeumËnh katanom† thc tuqa–ac metablht†c Xk, dojËntwn twnX0, . . . , Xk�1. EpomËnwc gia na perigràyoume thn Qn(⌘) thc sqËshc (1.2), gia kàje n 2 N0 kai kàje⌘ 2 Xn+1 kai àra gia na or–soume olÏklhrh thn oikogËneia katanom∏n peperasmËnhc diàstashc thcstoqastik†c diadikas–ac {Xn}n2N0 , arke– na perigràyoume

• thn katanom† thc X0 kai

• th desmeumËnh katanom† thc Xk, dojËntwn twn X0, . . . , Xk�1, gia kàje k 2 N.

H pr∏th plhrofor–a aforà thn arqik† katàstash tou sust†matoc, en∏ h de‘terh aforà th dunamik† tou,dhlad† ton nÏmo pou perigràfei thn exËlixh tou sust†matoc ston qrÏno. Parathr†ste thn analog–a meta aitiokratikà dunamikà sust†mata diakrito‘ qrÏnou, ta opo–a mporo‘me na perigràyoume d–nontac thnarqik† touc katàstash kai Ënan kanÏna gia to p∏c prok‘ptei h epÏmenh katàstash apÏ tic prohgo‘menec.

E–dame p∏c mporo‘me na perigràyoume pl†rwc tic statistikËc idiÏthtec miac stoqastik†c diadikas–acdiakrito‘ qrÏnou pou Ëqei †dh oriste– se kàpoion q∏ro pijanÏthtac. Arke– na xËroume thn arqik†thc katanom† kai th desmeumËnh katanom† thc epÏmenhc katàstashc, dojËntwn twn prohgo‘menwn. S'autÏ to shme–o ax–zei na anarwthjo‘me gia to ant–strofo er∏thma. E–nai pànta dunatÏ na bro‘meËnan katàllhlo q∏ro pijanÏthtac kai s' autÏn na or–soume mia stoqastik† diadikas–a {Xn}n2N0 , ∏steaut† na Ëqei mia sugkekrimËnh arqik† katanom† kai Ënan sugkekrimËno nÏmo exËlixhc me thn Ënnoiapou perigràyame parapànw; H apànthsh e–nai katafatik† kai bas–zetai sto Je∏rhma SunËpeiac touKolmogorov (Je∏rhma 3.5 sto [8]). H apÏdeixh Ïmwc auto‘ tou isqurismo‘ xefe‘gei apÏ touc skopo‘ctou maj†matoc. Gia thn ∏ra e–nai arketÏ na jumÏmaste Ïti to Je∏rhma SunËpeiac mac exasfal–zei pwcoi stoqastikËc diadikas–ec pou ja melet†soume e–nai uparktà majhmatikà antike–mena.

1.4 MarkobianËc alus–dec

'Otan jËloume na montelopoi†soume pragmatikà sust†mata e–nai pol‘ sunhjismËno o kanÏnac pou peri-gràfei th dunamik† tou sust†matoc na exartàtai mÏno apÏ thn trËqousa katàstash tou sust†matoc kaiÏqi apÏ to p∏c to s‘sthma brËjhke eke–. Ta stoqastikà sust†mata pou Ëqoun aut† thn idiÏthta qara-kthr–zontai wc markobianà. O anagn∏sthc e–nai pijanÏtata exoikeiwmËnoc me tËtoia parade–gmata miackai ta perissÏtera epitrapËzia paiqn–dia apotelo‘n markobianà sust†mata. Gia paràdeigma, an mpe–tes' Ëna dwmàtio Ïpou d‘o f–loi sac pa–zoun tàbli, gia na ektim†sete thn pijanÏthta n–khc kàje pa–kth,arke– na de–te thn trËqousa diamÏrfwsh twn pouli∏n. An xËrete aut† th diamÏrfwsh, h gn∏sh tou tiprohg†jhke den ja prosfËrei t–pota sthn prÏbley† sac gia thn exËlixh thc part–dac.

OrismÏc: Mia stoqastik† diadikas–a {Xn}n2N0 lËgetai markobian† alus–da (Markov chain), an, giakàje n 2 N, h desmeumËnh katanom† thc Xn+1 dojËntwn twn (X0, . . . , Xn), taut–zetai me th desmeumËnhkatanom† thc Xn+1 me mÏnh doje–sa thn Xn. EpomËnwc, h {Xn}n2N0 me timËc se Ënan arijm†simo q∏rokatastàsewn X e–nai markobian† alus–da, an, gia kàje n 2 N kai kàje v0, . . . , vn�1, x, y 2 X, Ëqoume

P⇥Xn+1 = y

��X0 = v0, . . . , Xn�1 = vn�1, Xn = x⇤= P

⇥Xn+1 = y

��Xn = x⇤. (1.5)

6

Stic perissÏterec markobianËc alus–dec pou parousiàzoun endiafËron, o kanÏnac P⇥Xn+1 = ·

��Xn = ·⇤

pou perigràfei thn exËlixh thc alus–dac, den exartàtai apÏ th qronik† paràmetro n. LËme tÏte Ïti halus–da e–nai qronikà omoiogen†c (time homogeneous) kai or–zoume tic pijanÏthtec metàbashc (transitionprobabilities) thc alus–dac

p : X⇥ X ! [0, 1], me t‘po p(x, y) = P⇥Xn+1 = y

��Xn = x⇤. (1.6)

Epeid† stic shmei∏seic autËc ja asqolhjo‘me mÏno me qronikà omoiogene–c alus–dec, sto ex†c ja para-le–petai h dieukr–nhsh gia lÏgouc oikonom–ac.

H sullog† P = {p(x, y)}x,y2X onomàzetai p–nakac pijanot†twn metàbashc (transition matrix) thc a-lus–dac. H orolog–a proËrqetai apÏ thn per–ptwsh pou o q∏roc katastàsewn X e–nai peperasmËnoc,opÏte mporo‘me na perigràyoume m–a sunàrthsh me ped–o orismo‘ to X⇥ X wc Ëna tetragwnikÏ p–naka.SugkekrimËna, an X = {v1, . . . , vN}, Ëqoume

P =

0

BBB@

p(v1, v1) p(v1, v2) · · · p(v1, vN )p(v2, v1) p(v2, v2) · · · p(v2, vN )...

.... . .

...p(vN , v1) p(vN , v2) · · · p(vN , vN )

1

CCCA. (1.7)

Ac sumbol–zoume me ⇡0 thn arqik† katanom† thc alus–dac, dhlad†

⇡0 : X ! [0, 1], me t‘po ⇡0(x) = P⇥X0 = x

⇤. (1.8)

Wc s.m.p. thc tuqa–ac metablht†c X0, h ⇡0 ikanopoie– tic parakàtw sunj†kec

⇡0(x) � 0, gia kàje x 2 X kaiX

x2X⇡0(x) = 1. (1.9)

Ant–stoiqa, efÏson h p(x, ·) e–nai h desmeumËnh s.m.p. thc Xn+1 dedomËnou Ïti Xn = x, oi pijanÏthtecmetàbashc thc markobian†c alus–dac {Xn}n2N0 ikanopoio‘n tic sunj†kec

p(x, y) � 0, gia kàje x, y 2 X kaiX

y2Xp(x, y) = 1, gia kàje x 2 X. (1.10)

OrismÏcJa lËme Ïti o P = {p(x, y)}x,y2X e–nai stoqastikÏc p–nakac (stochastic matrix), an ta stoiqe–atou ikanopoio‘n tic sunj†kec thc (1.10).

S‘mfwna me Ïsa e–dame sthn paràgrafo 1.3, h s.m.p. ⇡0 thc (1.8) kai o stoqastikÏc p–nakac P ={p(x, y)}x,y2X thc (1.6) kajor–zoun pl†rwc tic statistikËc idiÏthtec thc markobian†c alus–dac {Xn}n2N0 .Pràgmati, oi katanomËc peperasmËnhc diàstashc thc {Xn}n2N0 mporo‘n na upologisto‘n me thn bo†jeiathc (1.4). Gia kàje n 2 N0 kai kàje (x0, . . . , xn) 2 Xn+1 Ëqoume

P⇥X0 = x0, . . . , Xn = xn

⇤= ⇡0(x0) p(x0, x1) p(x1, x2) · · · p(xn�1, xn). (1.11)

Sta epÏmena kefàlaia ja do‘me Ïti gia na anal‘soume th sumperiforà miac alus–dac arke– na gnwr–zoumeautËc tic posÏthtec. 'Opwc kai sthn per–ptwsh twn pragmatik∏n tuqa–wn metablht∏n pou h katanom†touc mac e–nai gnwst†, o deigmatikÏc q∏roc ⌦ ston opo–o Ëqei oriste– h alus–da kai o t‘poc thc apei-kÏnishc ! 7! X·(!) den ja pa–xoun kanËna rÏlo.

Ja do‘me sta parade–gmata pou akoloujo‘n Ïti, sth montelopo–hsh pragmatik∏n susthmàtwn apÏ mar-kobianËc alus–dec, sun†jwc to prÏblhma upodeikn‘ei thn arqik† katanom† ⇡0 kai ton p–naka pijanot†twnmetàbashc P pou jËloume na Ëqei h alus–da h opo–a perigràfei to s‘sthma. To Je∏rhma SunËpeiac

7

tou Kolmogorov mac exasfal–zei Ïti, gia kàje s.m.p. ⇡0 kai gia kàje stoqastikÏ p–naka P , mporo‘mena kataskeuàsoume mia markobian† alus–da {Xn}n2N0 , h opo–a Ëqei thn ⇡0 wc arqik† katanom† kai tonP wc p–naka pijanot†twn metàbashc. Oi leptomËreiec aut†c thc kataskeu†c e–nai adiàforec gia thn a-nàlush tou montËlou, afo‘ h ⇡0 kai o P , Ïpwc e–pame, kajor–zoun tic statistikËc idiÏthtec thc alus–dac.

A B C

DE

Paràdeigma 4 'Ena Ëntomo kine–tai stic korufËc tou dipla-no‘ gràfou. Xekinà apÏ thn koruf† A. Se kàje b†ma, anbr–sketai sthn koruf† x, epilËgei tuqa–a m–a apÏ tic korufËcpou sundËontai me thn x mËsw miac akm†c tou gràfou kai me-taba–nei eke–.

Mporo‘me na perigràyoume thn k–nhsh tou entÏmou wc mia markobian† alus–da ston q∏ro katastàsewnX = {A,B,C,D,E}. EfÏson h alus–da xekinà apÏ thn koruf† A, h arqik† katanom† thc Ëqei s.m.p.

⇡0(x) = P⇥X0 = x

⇤= �A(x) =

(1, an x = A

0, an x 6= A.

Mporo‘me na perigràyoume thn arqik† katanom† kai wc Ëna diànusma gramm†

⇡0 =�⇡0(A),⇡0(B),⇡0(C),⇡0(D),⇡0(E)

�= (1, 0, 0, 0, 0).

O p–nakac pijanot†twn metàbashc P thc alus–dac e–nai o

P =

0

BBBB@

0 1/2 0 0 1/21/4 0 1/4 1/4 1/40 1/2 0 1/2 00 1/3 1/3 0 1/31/3 1/3 0 1/3 0

1

CCCCA.

Kàje tou gramm† perigràfei tic metabàseic apÏ mia koruf†. Ac do‘me gia paràdeigma p∏c prok‘ptei hpr∏th gramm† tou P , pou perigràfei tic metabàseic apÏ thn koruf† A. ApÏ thn A mporo‘me na pàmestic korufËc B † E me pijanÏthta 1/2 se kàje m–a. 'Ara h pr∏th gramm† tou P e–nai

�p(A,A), p(A,B), p(A,C), p(A,D), p(A,E)

�= (0, 1/2, 0, 0, 1/2).

Paràdeigma 5 Kànoume Ënan tuqa–o per–pato stouc akera–ouc xekin∏ntac apÏ to 0. Se kàje macb†ma str–boume Ëna kËrma pou Ëqei pijanÏthta p 2 (0, 1) na fËrei kefal† kàje forà pou to str–boume,anexàrthta apÏ tic àllec forËc. An to apotËlesma e–nai kefal†, metatopizÏmaste m–a jËsh dexià. An toapotËlesma e–nai gràmmata, metatopizÏmaste m–a jËsh aristerà. Mporo‘me na perigràyoume autÏn tonper–pato wc mia markobian† alus–da ston q∏ro katastàsewn Z. EfÏson xekinàme apÏ to 0, h arqik†katanom† Ëqei s.m.p.

⇡0(x) = P⇥X0 = x

⇤= �0(x) =

(1, an x = 0

0, an x 6= 0.

Oi pijanÏthtec metàbashc thc alus–dac, d–nontai apÏ thn

p(x, y) = P⇥Xn+1 = y

��Xn = x⇤=

8><

>:

p, an y = x+ 1

1� p, an y = x� 1

0, an |y � x| 6= 1

, x, y 2 Z.

8

Paràdeigma 6 Str–boume Ëna t–mio kËrma mËqri na emfaniste– gia pr∏th forà h akolouj–a apotele-smàtwn KGK. Mia pr∏th skËyh ja †tan na perigràyoume autÏ to pe–rama wc mia markobian† alus–da,pou h katàstas† thc e–nai to apotËlesma twn tri∏n teleuta–wn striyimàtwn. Me aut† thn epilog† oq∏roc katastàsewn ja †tan o X1 = {K,�}3. H arqik† katanom† thc alus–dac ja prok‘yei apÏ toapotËlesma twn tri∏n pr∏twn striyimàtwn. EfÏson to kËrma e–nai t–mio, ja prËpei na pàroume sans.m.p. thc X0 thn ⇡0(x) = 1/8, gia kàje x 2 X1. E–nai e‘kolo na perigràyoume kai tic pijanÏthtecmetàbashc thc alus–dac. Gia paràdeigma, apÏ thn katàstash KGG h alus–da mpore– na metabe– e–te sthnGGK me pijanÏthta 1/2 (an fËroume kor∏na) e–te sthn GGG me pijanÏthta 1/2 (an fËroume gràmmata).Epeid† to pe–rama telei∏nei Ïtan h alus–da ftàsei sthn KGK, mporo‘me na jewr†soume Ïti apÏ thn KGKparamËnoume sthn –dia katàstash me pijanÏthta 1. Mpore–te na sumplhr∏sete ton p–naka pijanot†twnmetàbashc gia exàskhsh.

Mporo‘me Ïmwc na perigràyoume to pe–rama kai pio oikonomikà, jewr∏ntac san katastàsh thc alus–dacpou to perigràfei to apotËlesma twn d‘o teleuta–wn striyimàtwn kai prosjËtontac mia katàstash sthnopo–a h alus–da ja metaba–nei, Ïtan to pe–rama telei∏sei. Me aut† thn epilog† o q∏roc katastàsewn thcalus–dac e–nai o X = {KK,K�,�K,��, T}, Ïpou h katàstash T kwdikopoie– to tËloc tou peiràmatoc.H arqik† katanom† thc alus–dac ja prok‘yei t∏ra apÏ to apotËlesma twn d‘o pr∏twn striyimàtwn.EfÏson to kËrma e–nai t–mio, h alus–da arqikà ja breje– se m–a apÏ tic KK, KG, GK † GG me pijanÏthta1/4 sthn kàje m–a. Mporo‘me loipÏn na perigràyoume thn arqik† katanom† thc alus–dac wc Ëna diànusmagramm†

⇡0 =�⇡0(KK), ⇡0(K�), ⇡0(�K), ⇡0(��), ⇡0(T )

�= (1/4, 1/4, 1/4, 1/4, 0).

O p–nakac pijanot†twn metàbashc thc alus–dac e–nai o

P =

0

BBBB@

1/2 1/2 0 0 00 0 0 1/2 1/21/2 1/2 0 0 00 0 1/2 1/2 00 0 0 0 1

1

CCCCA.

Paràdeigma 7 To epitrapËzio paiqn–di fidàki anàmesa se d‘o pa–ktec mpore– na perigrafe– wc miamarkobian† alus–da ston q∏ro katastàsewn

X = {1, 2, . . . , 100}⇥ {1, 2, . . . , 100}⇥ {A,B}.

To paiqn–di br–sketai sthn katàstash (m,n, b), an to piÏni tou pa–kth A br–sketai sto tetràgwno m,to piÏni tou pa–kth B br–sketai sto tetràgwno n kai e–nai h seirà tou pa–kth b na r–xei to zàri. Harqik† katanom† thc alus–dac, an o pa–kthc A pa–zei pr∏toc, e–nai h ⇡0(x) = �(1,1,A)(x). Ja †tankopi∏dec na perigràyoume ton p–naka pijanot†twn metàbashc aut†c thc alus–dac san Ëna tetragwnikÏ20.000⇥ 20.000 p–naka, allà oi pijanÏthtec metàbashc mporo‘n na perigrafo‘n saf∏c. Gia paràdeigmap�(1, 1, A), (3, 1, B)

�= 1/6, afo‘ to paiqn–di ja metabe– apÏ thn katàstash (1, 1, A) sthn katàstash

(3, 1, B), an h zarià tou pa–kth A e–nai 2.

Paràdeigma 8 'Enac mpasketmpol–stac propone–tai stic ele‘jerec bolËc. Kàje forà pou ektele– miabol†, h pijanÏthta na eustoq†sei e–nai 8/10, an Ëqei eustoq†sei kai stic d‘o prohgo‘menec prospàjeiËctou, 5/10, an Ëqei astoq†sei kai stic d‘o prohgo‘menec prospàjeiec kai 7/10 se kàje àllh per–ptwsh.Or–zoume th stoqastik† diadikas–a {Xn}n2N, Ïpou Xn = 1, an h n-ost† tou prospàjeia e–nai e‘stoqhkai Xn = 0, an h n-ost† tou prospàjeia e–nai àstoqh. E–nai h {Xn}n2N markobian† alus–da;

Diaisjhtikà perimËnei kane–c Ïti h apànthsh e–nai arnhtik†. H tim† thc Xn Ëqei thn plhrofor–a mÏno giato apotËlesma thc teleuta–ac bol†c, en∏ h pijanÏthta eustoq–ac sthn epÏmenh prospàjeia exartàtai

9

apÏ to apotËlesma twn d‘o teleuta–wn bol∏n. Gia na apode–xoume austhrà Ïti h {Xn}n2N den e–naimarkobian†, arke– na parathr†soume Ïti

P⇥Xn+1 = 1

��Xn�1 = Xn = 1⇤= 8/10 6= 7/10 = P

⇥Xn+1 = 1

��Xn�1 = 0, Xn = 1⇤.

An Ïmwc h {Xn}n2N †tan markobian†, oi d‘o pijanÏthtec sthn parapànw sqËsh ja †tan –sec, afo‘ kaioi d‘o ja sunËpiptan me thn P

⇥Xn+1 = 1

��Xn = 1⇤.

To prÏblhma ed∏ e–nai Ïti o q∏roc katastàsewn thc alus–dac X = {0, 1} e–nai pol‘ mikrÏc gia nakwdikopoi†sei thn plhrofor–a pou qreiazÏmaste gia na perigràyoume thn katanom† thc epÏmenhc ka-tàstashc. Mporo‘me Ïmwc, megal∏nontac ton q∏ro katastàsewn, na perigràyoume thn propÏnhsh touajlht† wc mia markobian† alus–da.

An gia paràdeigma or–soume th stoqastik† diadikas–a {Yn}n2N, me Yn = (Xn, Xn+1), aut† e–nai miamarkobian† alus–da ston q∏ro katastàsewn X0 = {(0, 0), (0, 1), (1, 0), (1, 1)} me p–naka pijanot†twnmetàbashc

P =

0

BB@

1/2 1/2 0 00 0 3/10 7/10

3/10 7/10 0 00 0 2/10 8/10

1

CCA . 2

Sunhj–zoume na lËme Ïti gia mia markobian† alus–da pareljÏn kai mËllon e–nai anexàrthta, me dedomËnoto parÏn. H markobian† idiÏthta (Markov property) e–nai h majhmatik† diat‘pwsh aut†c thc Ëkfrashc.

Je∏rhma 1 (markobian† idiÏthta) 'Estw {Xn}n2N0 markobian† alus–da se Ëna arijm†simo q∏ro ka-tastàsewn X me p–naka pijanot†twn metàbashc P . Gia k 2 N0, or–zoume th stoqastik† diadikas–a{Yn}n2N0 me t‘po Yn = Xn+k. DedomËnou Ïti Xk = x 2 X, h diadikas–a {Yn}n2N0 e–nai markobian†alus–da ston X, me arqik† katanom† �x, p–naka pijanot†twn metàbashc P kai e–nai anexàrthth apÏ ticX0, X1, . . . , Xk.

ApÏdeixh: Arke– na de–xoume Ïti, dedomËnou Ïti Xk = x, gia kàje n 2 N oi (Y0, Y1, . . . , Yn) e–nai anexàr-thtec apÏ tic X0, X1, . . . , Xk kai h apÏ koino‘ touc katanom† d–netai apÏ thn (1.11) me arqik† katanom†�x. Arke– epomËnwc na de–xoume Ïti, gia kàje y = (y0, . . . , yn) 2 Xn+1 kai kàje z = (z0, . . . , zk) 2 Xk+1

P⇥{(Y0, . . . , Yn) = y} \A(z)

��Xk = x⇤= �x(y0)p(y0, y1) · · · p(yn�1, yn)P

⇥A(z)

��Xk = x⇤,

Ïpou A(z) = {(X0, . . . , Xk) = z}. Pràgmati Ïmwc,

P⇥{(Y0, . . . , Yn) = y} \A(z)

��Xk = x⇤= �x(y0)�x(zk)

P⇥X0 = z0, . . . , Xk = x,Xk+1 = y1, . . . , Xn+k = yn

⇤

P⇥Xk = x

⇤

=�x(zk)⇡0(z0)p(z0, z1) · · · p(zk�1, x)

P⇥Xk = x

⇤ �x(y0)p(x, y1) · · · p(yn�1, yn)

= P⇥A(z)

��Xk = x⇤�x(y0)p(y0, y1) · · · p(yn�1, yn). 2

BlËpoume loipÏn Ïti, an gnwr–zoume thn paro‘sa jËsh thc alus–dac, dedomËnou dhlad† Ïti Xk = x, tÏteto mËllon thc alus–dac, to opo–o ekfràzetai apÏ thn alus–da {Yn}n2N0 , e–nai anexàrthto tou pareljÏntocthc, dhlad† twn tuqa–wn metablht∏n X0, . . . , Xk kai Ëqei màlista tic –diec statistikËc idiÏthtec Ïpwc miaalus–da pou xekinà apÏ to x me tic –diec pijanÏthtec metàbashc, P . 2

'Otan h alus–da {Xn}n2N0 xekinà apÏ thn katàstash x 2 X, ja qrhsimopoio‘me ton sumbolismÏ Px

⇥A⇤

10

gia thn pijanÏthta enÏc endeqomËnou A pou exartàtai apÏ thn troqià thc alus–dac. Gia paràdeigma, anv = (v0, . . . , vn) 2 Xn+1, Ëqoume

Px

⇥(X0, . . . , Xn) = v

⇤= �x(v0)p(v0, v1) · · · p(vn�1, vn).

Qrhsimopoi∏ntac th markobian† idiÏthta gia k = 0, Ëqoume Ïti P⇥·��X0 = x

⇤= Px

⇥·⇤.

1.5 Ask†seic

'Askhsh 1 Jewr†ste d‘o anexàrthtec tuqa–ec metablhtËc A,⇥ kai or–ste th stoqastik† diadikas–a{Xt}t�0 me t‘po

Xt = A sin(!t+⇥),

Ïpou ! 2 R. An h A akolouje– ekjetik† katanom† me rujmÏ � = 1 kai h ⇥ akolouje– omoiÏmorfhkatanom† sto [0, 2⇡], upolog–ste tic E

⇥Xt

⇤kai E

⇥XtXs

⇤, gia kàje s, t � 0.

'Askhsh 2 An {Xt}t�0 e–nai h stoqastik† diadikas–a tou Parade–gmatoc 3, or–zoume

Yt = e�tXe2t .

De–xte Ïti h {Yt}t�0 e–nai diadikas–a Gauss, me m(t) = E⇥Yt⇤= 0 kai ⇢(s, t) = Cov(Yt, Ys) = e�|t�s|.

'Askhsh 3 Jewr†ste mia diadikas–a Gauss {Xt}t�0, me m(t) = E⇥Xt

⇤= 0, gia kàje t � 0 kai

⇢(s, t) = Cov�Xt, Xs) =

1

2

�t2H + s2H � |t� s|2H

�, s, t � 0,

gia kàpoio H 2 (0, 1).a) De–xte Ïti gia kàje t � 0 Ëqoume Xt ⇠ N

�0, t2H

�.

b) Upolog–ste gia t � 0 kai h > 0 th mËsh tim† kai th diasporà thc prosa‘xhshc Xt+h �Xt kai de–xteÏti h Xt+h �Xt akolouje– thn –dia katanom† me thn Xh.g) De–xte Ïti gia kàje t, h > 0, h Xt+h �Xt e–nai anexàrthth apÏ thn Xt, an kai mÏno an H = 1/2.d) Poia gnwst† mac stoqastik† diadikas–a e–nai h {Xt}t�0 Ïtan H = 1/2;e) Or–zoume gia kàje t � 0 : Yt = ↵�HX↵t, Ïpou ↵ jetikÏc pragmatikÏc arijmÏc. De–xte Ïti E

⇥Yt⇤=

E⇥Xt

⇤, gia kàje t � 0 kai Cov

�Yt, Ys) = Cov

�Xt, Xs), gia kàje s, t � 0. P∏c ermhne‘ete autÏ to

apotËlesma;

'Askhsh 4 a) Antikatast†ste ta s‘mbola * me arijmo‘c, ∏ste o p–nakac

P =

0

BBBB@

0 3/4 ⇤ 0 03/4 0 0 1/8 ⇤1/2 1/4 1/4 ⇤ ⇤⇤ 3/5 1/5 1/5 ⇤0 0 1/10 1/5 ⇤

1

CCCCA

na e–nai p–nakac pijanot†twn metàbashc miac markobian†c alus–dac {Xn}n2N0 ston q∏ro katastàsewnX = {1, 2, . . . , 7} me pijanÏthtec metàbashc p(i, j) gia kàje i, j 2 X.

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'Askhsh 5 Sto diplanÏ sq†ma fa–netai h kàtoyh enÏc spitio‘ me pËntedwmàtia: kouz–na (K), biblioj†kh (B), salÏni (S), upnodwmàtio (U) kai mpànio(M), kaj∏c kai oi pÏrtec pou ta sundËoun. 'Ena Ëntomo pou zei sto sp–ti kàjebràdu diasq–zei tuqa–a m–a apÏ tic pÏrtec tou dwmat–ou sto opo–o br–sketaikai paramËnei sto dwmàtio pou odhge– h pÏrta mËqri to epÏmeno bràdu. Arqikàto Ëntomo br–sketai sto mpànio. An {Xn}n2N0 e–nai h markobian† alus–daston q∏ro katastàsewn X ={K, B, S, U, M} pou perigràfei th jËsh touentÏmou metà apÏ n bràdua, bre–te ton p–naka pijanot†twn metàbas†c thc.

'Askhsh 6 'Eqoume pËnte qartià thc tràpoulac, ta tËssera e–nai eptà ko‘pa kai to Ëna r†gac spaj–.Ta apl∏noume sth seirà se Ëna trapËzi kai se kàje b†ma epilËgoume Ëna apÏ ta d‘o akra–a qartià (toaristerÏtero me pijanÏthta 2/3, to dexiÏtero me pijanÏthta 1/3) kai to topojeto‘me sth mËsh. Kata-skeuàste Ënan katàllhlo q∏ro katastàsewn kai tic ant–stoiqec pijanÏthtec metàbashc miac markobian†calus–dac pou ja periËgrafe thn jËsh tou r†ga.

'Askhsh 7 R–qnoume Ëna zàri mËqri na fËroume pËnte suneqÏmenec forËc Ëxi. Perigràyte mia marko-bian† alus–da ston q∏ro katastàsewn X = {0, 1, 2, 3, 4, 5} pou ja montelopoio‘se autÏ to paiqn–di.Kànte to –dio gia thn per–ptwsh sthn opo–a r–qnoume to zàri mËqri na emfaniste– h akolouj–a 65656.

'Askhsh 8 Epanalambànoume r–yeic enÏc zario‘ kai gia n 2 N sumbol–zoume me Sn to àjroisma twn npr∏twn zari∏n mac. An Xn = Sn(mod5), de–xte Ïti h {Xn}n2N e–nai mia markobian† alus–da kai bre–teton p–naka pijanot†twn metàbas†c thc.

'Askhsh 9 Aut† h àskhsh mac didàskei p∏c na prosomoi∏soume mia katanom† ston X me th bo†jeiamiac genn†triac tuqa–wn arijm∏n. 'Estw X = {v1, v2, . . .} Ënac arijm†simoc q∏roc katastàsewn. Gia mias.m.p. {pi}i2N ston X or–zoume th sunàrthsh:

�p : (0, 1] ! X, �p(x) = vk, ank�1X

j=1

pi < x kX

j=1

pi.

De–xte Ïti, an h tuqa–a metablht† U Ëqei omoiÏmorfh katanom† sto [0,1], tÏte h �p(U) Ëqei s.m.p.{pi}i2N.

'Askhsh 10 Jewr†ste Ënan arijm†simo q∏ro X kai mia sunàrthsh � : X ⇥ [0, 1] ! X. An {Un}n2Ne–nai mia akolouj–a apÏ anexàrthtec, isÏnomec tuqa–ec metablhtËc, me omoiÏmorfh katanom† sto [0,1] kaior–soume anadromikà th stoqastik† diadikas–a {Xn}n2N0 wc

X0 = x 2 X, Xn = �(Xn�1, Un) gia n 2 N,

tÏte h {Xn}n2N0 e–nai markobian† alus–da. Poiec e–nai oi pijanÏthtec metàbashc aut†c thc alus–dac; Meth bo†jeia kai thc prohgo‘menhc àskhshc, exhg†ste p∏c mporo‘me na prosomoi∏soume mia markobian†alus–da me dedomËnec pijanÏthtec metàbashc.

'Askhsh 11 S' Ëna ràfi thc biblioj†khc sac upàrqoun tr–a bibl–a: Algebra, Basic Topology, Calculus,pou ja sumbol–zoume me A,B,C gia suntom–a. Kàje prw– pa–rnete tuqa–a Ëna bibl–o apÏ th jËsh toume pijanÏthtec p, q, r, ant–stoiqa. 'Otan telei∏nete to diàbasmà sac gia thn hmËra, to xanabàzete storàfi sthn aristerÏterh jËsh. H diàtaxh twn bibl–wn e–nai mia markobian† alus–da ston q∏ro X twnmetajËsewn twn sumbÏlwn {A,B,C}. Poioc e–nai o p–nakac pijanot†twn metàbashc aut†c thc alus–dac;

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'Askhsh 12 Sto montËlo diàqushc tou Ehrenfest, N swmat–dia topojeto‘ntai se Ëna doqe–o me d‘odiamer–smata, A kai B. Se kàje b†ma epilËgoume tuqa–a Ëna apÏ ta N swmat–dia kai tou allàzoumediamËrisma. 'Estw {Xn}n2N0 h markobian† alus–da ston q∏ro katastàsewn X = {0, 1, . . . , N} pouperigràfei to pl†joc twn swmatid–wn sto diamËrisma A metà apÏ n b†mata. Poiec e–nai oi pijanÏthtecmetàbashc thc {Xn}n;

'Askhsh 13 Montelopoi†ste to paiqn–di tou parade–gmatoc 6 wc mia markobian† alus–da se Ënan q∏roX me 4 mÏno katastàseic.

'Askhsh 14 'Enac pantop∏lhc efodiàzetai kàje prw– me d‘o pakËta mpiskÏta Alfajor. H hmer†siaz†thsh gia ta mpiskÏta autà e–nai mia tuqa–a metablht† pou akolouje– gewmetrik† katanom† me paràmetrop = 1/2. An qjec bràdu den e–qan me–nei kajÏlou mpiskÏta Alfajor kai {Xn}n2N e–nai to pl†joctwn pakËtwn pou upàrqei sto pantopwle–o to bràdu thc n-ost†c hmËrac, de–xte Ïti h {Xn}n2N e–naimarkobian† alus–da kai bre–te tic pijanÏthtec metàbas†c thc.

'Askhsh 15 Jewr†ste d‘o akolouj–ec {Xn, Yn}0nN�1 apÏ anexàrthta, tuqa–a, dekadikà yhf–a.Sqhmat–ste touc akera–ouc

X = X0 +X1 ⇥ 10 + · · ·+XN�1 ⇥ 10N�1, Y = Y0 + Y1 ⇥ 10 + · · ·+ YN�1 ⇥ 10N�1

kai prosjËste touc, Ïpwc màjame sto dhmotikÏ, xekin∏ntac apÏ ta yhf–a twn monàdwn X0, Y0, suneq–zo-ntac me ta yhf–a twn dekàdwnX1, Y1 k.lp., metafËrontac to krato‘meno, Ïpou qreiàzetai. An Cn 2 {0, 1}e–nai to krato‘meno thc prÏsjeshc pou metafËroume sto n-ostÏ b†ma, de–xte Ïti h {Cn}0nN�1 e–naimia markobian† alus–da ston q∏ro katastàsewn X = {0, 1} kai bre–te ton p–naka pijanot†twn metàbashcthc alus–dac.

1.6 Arijmhtikà peiràmata

'Askhsh 16 Katebàste kai egkatast†ste th gl∏ssa Python 2.7.7. Mpore–te na bre–te ta sqetikàarqe–a ed∏. Anàloga me to leitourgikÏ s‘sthma tou H/U sac, mpore–te na bre–te kai oloklhrwmËnapakËta. An o upologist†c sac den trËqei to leitourgikÏ s‘sthma Linux, pijanà na bre–te qr†simo naegkatast†sete Ënan exomoiwt†, s‘mfwna me tic odhg–ec pou ja bre–te ed∏.

'Askhsh 17 Katebàste to prÏgramma simple markov chain lib.py kai apojhke‘ste to ston ka-tàlogo pou ja doulËyete. S' aut† thn fàsh den qreiàzetai kan na to ano–xete. To prÏgramma autÏulopoie– ton algÏrijmo thc 'Askhshc 10. Ja to qrhsimopoio‘me san biblioj†kh sta epÏmena progràm-mata pou ja ftiàxoume.

'Askhsh 18 Katebàste kai trËxte to prÏgramma test.py. To prÏgramma autÏ prosomoi∏nei ta pr∏tadËka b†mata miac alus–dac pou kine–tai ston q∏ro katastàsewn X = {1, 2, 3, 4, 5} me p–naka pijanot†twnmetàbashc

P =

0

BBBB@

0 1/2 1/2 0 01/3 0 0 2/3 00 0 1 0 01/2 0 0 1/2 00 0 0 0 1

1

CCCCA

xekin∏ntac apÏ thn katàstash 1. TrËxte to prÏgramma merikËc forËc kai sth sunËqeia ftiàxte ËnaprÏgramma pou prosomoi∏nei ta e–kosi pr∏ta b†mata thc 'Askhshc 11.

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