Kefàlaio 6

Asumptwtikà Jewr†mata

6.1 Eisagwg†

Sto kefàlaio autÏ ja melet†soume thn asumptwtik† sumperiforà markobian∏n alus–dwn, to p∏c dh-lad† autËc sumperifËrontai se bàjoc qrÏnou. H katanÏhsh thc asumptwtik†c sumperiforàc touc e–naishmantik† gia d‘o kur–wc lÏgouc. AfenÏc, upàrqoun sust†mata pou montelopoio‘ntai apÏ markobianËcalus–dec kai ta opo–a, afo‘ peràsoun mia parodik† fàsh, telikà leitourgo‘n se mia katàstash isorro-p–ac. AfetËrou, an jËloume na qrhsimopoi†soume idËec apÏ tic markobianËc alus–dec gia na ftiàxoumeupologistiko‘c algor–jmouc, e–nai shmantikÏ na xËroume th sumperiforà tou algor–jmou metà apÏ pol-lËc epanal†yeic, na mporo‘me na ektim†soume pÏso gr†gora sukl–nei kai p∏c exartàtai to sfàlma thcprosËggishc apÏ to pl†joc twn epanal†yewn pou ja ektelËsoume. ParÏmoio ulikÏ mpore–te na bre–testic anaforËc [7] kai [6].

6.2 PeriodikÏthta

Ac jewr†soume mia alus–da pou kine–tai stic korufËc enÏc tetrag∏nou ABGD. Xekin∏ntac apÏ thnkoruf† A, se kàje b†ma metaba–nei se m–a apÏ tic d‘o geitonikËc thc korufËc me pijanÏthta 1/2. E‘kolablËpei kane–c Ïti h monadik† anallo–wth katanom† aut†c thc alus–dac e–nai h omoiÏmorfh katanom†⇡ = (1/4, 1/4, 1/4, 1/4) kai Ïpwc e–dame sto prohgo‘meno kefàlaio autÏ e–nai mÏno upoy†fio Ïrio thckatanom†c ⇡n thc alus–dac metà apÏ n b†mata. Den e–nai d‘skolo na dei kane–c Ïmwc Ïti ⇡n 6! ⇡.Pràgmati, efÏson h alus–da xekinà apÏ to A, ja br–sketai opwsd†pote sto B † sto D metà apÏ perittÏarijmÏ bhmàtwn kai sto A † sto G metà apÏ àrtio arijmÏ bhmàtwn. 'Etsi, p.q.

⇡2n+1(A) = P⇥X2n+1 = A |X0 = A

⇤= 0 6! 1/4.

BlËpoume loipÏn Ïti sto paràdeigmà mac upàrqoun upos‘nola tou q∏rou katastàsewn pou e–nai pro-sbàsima mÏno se sugkekrimËnec qronikËc stigmËc. Aut† h idiÏthta apotele– katà kàpoion trÏpo miapajolog–a pou empod–zei th s‘gklish thc katanom†c thc alus–dac ⇡n, kai se aut† thn paràgrafo japrospaj†soume na thn katano†soume.

OrismÏc: Gia mia katàstash x 2 X or–zoume to s‘nolo twn dunat∏n qrÏnwn epistrof†c sto x

R(x) = {n 2 N : pn(x, x) > 0}, Ïpou jum–zoume p(n)(x, x) = P⇥Xn = x |X0 = x

⇤.

Ja onomàzoume ton mËgisto koinÏ diairËth tou sunÏlou R(x) per–odo thc katàstashc x kai ja ton sum-bol–zoume me d(x). Sthn eidik† per–ptwsh pou d(x) = 1, ja lËme Ïti h katàstash x e–nai aperiodik†.

92

Paràdeigma 47 Sto diplanÏ sq†ma fa–nontai oi kata-stàseic kai oi dunatËc metabàseic miac markobian†c alus–dacpou kine–tai s' Ënan q∏ro me okt∏ katastàseic. Mia prosana-tolismËnh akm† apÏ to x sto y ston diplanÏ gràfo antistoiqe–se jetik† pijanÏthta metàbashc p(x, y). Oi upÏloipec meta-bàseic Ëqoun mhdenik† pijanÏthta. 'Eqoume R(1) = R(2) =R(3) = R(6) = {2, 3, . . .}, R(4) = {3, 5, 6, 7, 8, . . .}, R(5) =N, R(7) = R(8) = {2, 4, 5, 6, . . .}. Sto paràdeigma autÏ Ëqou-me d(x) = 1 gia Ïlec tic katastàseic x thc alus–dac, epomËnwcÏlec oi katastàseic e–nai aperiodikËc.

1

3

2

5

4

7

6

8

Parathr†ste Ïti, gia kàje x 2 X, to s‘nolo R(x) e–nai Ëna upos‘nolo tou N pou e–nai kleistÏ wc procthn prÏsjesh, dhlad†

m,n 2 R(x) ) m+ n 2 R(x).

Diaisjhtikà autÏ shma–nei Ïti, an e–nai dunatÏn na epistrËyoume sto x metà apÏ m b†mata, allà kai metàapÏ n b†mata, tÏte e–nai dunatÏn na epistrËyoume sto x kai metà apÏ m+ n b†mata. Pràgmati, apÏ ticexis∏seic Chapman-Kolmogorov, Ëqoume

p(m+n)(x, x) =X

y2Xp(m)(x, y)p(n)(y, x) � p(m)(x, x)p(n)(x, x) > 0,

efÏson m,n 2 R(x). Gia tËtoia s‘nola Ëqoume to akÏloujo l†mma.

L†mma 8 An Ëna s‘nolo R ⇢ N e–nai kleistÏ wc proc thn prÏsjesh kai Ëqei mËgisto koinÏ diairËthd 2 N, tÏte upàrqei n0 2 N tËtoio ∏ste {n0d, (n0 + 1)d, (n0 + 2)d, . . .} ⇢ R. To R periËqei epomËnwctelikà Ïla ta pollaplàsia tou d.

ApÏdeixh: Ja sumbol–zoume me dN to s‘nolo twn fusik∏n pou e–nai pollaplàsia tou d, dhlad†dN = {d, 2d, 3d, . . .}. An d 2 R, efÏson to R e–nai Ëna kleistÏ wc proc thn prÏsjesh upos‘nolo tou N,Ëqoume R = dN. An pàli d /2 R, tÏte, efÏson to d e–nai o mËgistoc koinÏc diairËthc tou R, ja upàrqounm1, . . . ,mk 2 R kai p1, p2, . . . , pk 2 Z \ {0} tËtoia ∏ste

kX

i=1

pimi = d. (6.1)

'Estw I ⇢ {1, 2, . . . , k} to s‘nolo twn deikt∏n i gia touc opo–ouc pi > 0 kai I 0 = {1, 2, . . . , k} \ I tos‘nolo twn deikt∏n i gia touc opo–ouc pi < 0. Profan∏c, to I den mpore– na e–nai kenÏ afo‘ to dex–mËloc thc (6.1) e–nai fusikÏc arijmÏc. O‘te to I 0 mpore– na e–nai kenÏ Ïmwc, afo‘ kàje prosjetËoc stoaristerÏ mËloc thc (6.1) e–nai megal‘teroc † –soc apÏ 2d kat' apÏluth tim† sthn per–ptwsh (d /2 R) pouexetàzoume. Mporo‘me epomËnwc na xanagràyoume thn (6.1) wc

X

i2Ipimi �

X

i2I0(�pi)mi = d.

EfÏson to R e–nai kleistÏ wc proc thn prÏsjesh kai pi > 0 gia i 2 I Ëqoume Ïti m =P

i2I pimi 2 R.Ant–stoiqa, Ëqoume Ïti n =

Pi2I0(�pi)mi 2 R. Br†kame epomËnwc d‘o stoiqe–a m,n 2 R tËtoia ∏ste

m = n+ d. Ja de–xoume Ïti to R periËqei kàje fusikÏ thc morf†c Nd gia Nd � n(n� 1). Pràgmati,an gràyoume thn tautÏthta thc akËraiac dia–reshc tou N me to n Ëqoume

N = qn+ �, gia kàpoia q � n� 1

d, � 2 {0, 1, 2, . . . , n� 1}. (6.2)

93

'Eqoume t∏raNd = qnd+ �d = qnd+ �(m� n) = �m+ (qd� �)n.

Oi � kai qd�� e–nai mh arnhtiko– akËraioi lÏgw thc (6.2) kai toulàqiston Ënac apÏ auto‘c e–nai diàforocapÏ to mhdËn afo‘ Ëqoun àjroisma qd. EfÏson m,n 2 R kai to R e–nai kleistÏ wc proc thn prÏsjesh,ja Ëqoume Nd 2 R gia kàje fusikÏ N � n(n� 1)/d.

Ja qrhsimopoi†soume t∏ra to L†mma 8 gia na de–xoume Ïti h per–odoc e–nai qarakthristikÏ klàshc.

Je∏rhma 26 An d‘o katàstaseic an†koun sthn –dia klàsh epikoinwn–ac, tÏte Ëqoun thn –dia per–odo.

ApÏdeixh: An oi x, y 2 X epikoinwno‘n amf–droma, upàrqoun m,n 2 N tËtoia ∏ste

p(m)(x, y) > 0 kai p(n)(y, x) > 0.

EpiplËon, an ` 2 R(x), tÏte ja Ëqoume m+ n+ ` 2 R(y). Pràgmati,

p(m+n+`)(y, y) =X

z,w2Xp(n)(y, z)p(`)(z, w)p(m)(w, y) � p(n)(y, x)p(`)(x, x)p(m)(x, y) > 0.

ApÏ to L†mma 8, upàrqei kàpoio q 2 N tËtoio ∏ste qd(x), (q + 1)d(x) 2 R(x). ApÏ thn prohgo‘menhparat†rhsh, ja Ëqoume loipÏn

m+ n+ qd(x) 2 R(y) kai m+ n+ (q + 1)d(x) 2 R(y).

EpomËnwc, d(y)/m + n + qd(x) kai d(y)/m + n + (q + 1)d(x) kai àra d(y)/d(x). Enallàssontac tonrÏlo twn x, y sto parapànw epiqe–rhma, pa–rnoume Ïti d(x)/d(y) kai àra d(x) = d(y).

UpÏ to fwc tou Jewr†matoc 26 Ëqei nÏhma na milàme gia thn per–odo miac klàshc † gia thn per–odo miacmh upobibàsimhc alus–dac kai na qarakthr–zoume mia tËtoia alus–da aperiodik† Ïtan h per–odÏc Ïlwn twnkatastàse∏n thc e–nai 1. To akÏloujo pÏrisma e–nai suqnà qr†simo giat– d–nei mia ikan† sunj†kh ∏stena e–nai mia alus–da aperiodik† pou mpore– na elegqje– me apl† episkÏphsh.

PÏrisma 10 An mia mh upobibàsimh alus–da Ëqei mia katàstash x gia thn opo–a p(x, x) > 0, tÏte halus–da e–nai aperiodik†.

ApÏdeixh: EfÏson p(x, x) > 0 Ëqoume 1 2 R(x) kai àra d(x) = 1. Epeid† h alus–da e–nai mh upobi-bàsimh, Ïlec oi katastàseic thc ja Ëqoun thn –dia per–do me thn x, dhlad† 1.

To akÏloujo l†mma ja mac fane– ep–shc qr†simo stic epÏmenec paragràfouc.

L†mma 9 An h {Xn}n2N0 e–nai mia mh upobibàsimh kai aperiodik† alus–da se Ënan q∏ro katastàsewnX me pijanÏthtec metàbashc p(·, ·), tÏte gia kàje x, y 2 X upàrqei n0 2 N tËtoio ∏ste p(n)(x, y) > 0,gia kàje n � n0.

ApÏdeixh: EfÏson d(x) = 1 apÏ to L†mma 8 upàrqei Ëna n1 2 N tËtoio ∏ste p(n)(x, x) > 0, gia kàjen � n1. EfÏson h {Xn}n e–nai mh upobibàsimh upàrqei akÏmh Ëna m 2 N tËtoio, ∏ste p(m)(x, y) > 0.An t∏ra n � n1 +m, Ëqoume

p(n)(x, y) =X

z2Xp(n�m)(x, z)p(m)(z, y) � p(n�m)(x, x)p(m)(x, y) > 0.

94

ProsËxte Ïti sto paràdeigma pou e–dame sthn arq† thc paragràfou, gia to opo–o h katanom† ⇡n thcalus–dac metà apÏ n b†mata den sugkl–nei sthn anallo–wth katanom† thc alus–dac, h per–odoc thcalus–dac e–nai 2. O lÏgoc pou h s‘gklish apotugqànei e–nai giat– an mia mh upobibàsimh alus–da Ëqeiper–odo d, o q∏roc katàstasewn diamer–zetai se d s‘nola X = [d�1

j=0Cj , se kajËna apÏ ta opo–a h alus–dampore– na epistrËyei mÏno metà apÏ Ëna arijmÏ bhmàtwn pou e–nai pollaplàsio tou d. Dhlad†,

P⇥Xn 2 Cj |X0 2 Cj

⇤6= 0 , d/n.

Pio sugkekrimËna Ëqoume to akÏloujo je∏rhma, tou opo–ou h apÏdeixh af†netai san àskhsh.

Je∏rhma 27 'Estw {Xn}n2N0 mia mh upobibàsimh alus–da se Ënan q∏ro katastàsewn X 3 x mepijanÏthtec metàbashc p(·, ·) kai per–odo d. TÏte h alus–da {Yn}n2N0 me Yn = Xnd (dhlad† h alus–dapou prok‘ptei an deigmatolhpto‘me thn {Xn}n kàje d b†mata) e–nai upobibàsimh stic d aperiodikËcklàseic

Cv = {y 2 X : p(dn+v)(x, y) > 0, gia kàpoio n 2 N0}, v 2 0, 1, 2, . . . d� 1.

An h alus–da {Xn}n2N e–nai gnhs–wc epanalhptik† me anallo–wth katanom† ⇡, tÏte h anallo–wth kata-nom† ⇡v thc alus–dac {Yn}n2N0 pou sthr–zetai sthn klàsh Cv d–netai apÏ thn

⇡v(y) =

(d⇡(y), an y 2 Cv

0, an y /2 Cv.(6.3)

Sthn paràgrafo 6.4 ja do‘me Ïti an mia mh upobibàsimh kai gnhs–wc epanalhptik† alus–da e–nai aperio-dik†, tÏte se bàjoc qrÏnou h katanom† thc alus–dac ⇡n ja sugkl–nei sthn anallo–wth katanom† thc.AutÏ e–nai Ëna pol‘ qr†simo apotËlesma, giat– ektÏc tou Ïti perigràfei thn asumptwtik† sumperifo-rà thc alus–dac, prosfËrei kai Ënan upologistikÏ trÏpo gia na pàroume de–gmata apÏ mia katanom† ⇡.Qreiàzetai apl∏c na kataskeuàsoume mia mh upobibàsimh, gnhs–wc epanalhptik† kai aperiodik† alus–dapou Ëqei anallo–wth katanom† thn ⇡ kai na af†soume aut† thn alus–da na kànei arketà b†mata ∏ste hkatanom† thc na plhsiàsei thn ⇡. Aut† h idËa e–nai h bàsh thc mejÏdou Markov Chain Monte Carlo(MCMC), pou ja melet†soume ektenËstera sto Kefàlaio 7.

Gia thn apÏdeixh thc s‘gklishc ⇡n ! ⇡ ja qrhsimopoi†soume mia pol‘ isqur† pijanojewrhtik† teqni-k†, th s‘zeuxh (coupling). LÏgw thc qrhsimÏthtac aut†c thc teqnik†c ja afier∏soume thn epÏmenhparàgrafo sto na thn perigràyoume mËsa apÏ d‘o parade–gmata.

6.3 S‘zeuxh

H s‘zeuxh e–nai mia pol‘ genik† mËjodoc me thn opo–a mporo‘me na pàroume qr†sima apotelËsmata giathn katanom† tuqa–wn metablht∏n, orismËnwn en gËnei se diaforetiko‘c q∏rouc pijanÏthtac, jewr∏ntacËnan megal‘tero q∏ro pijanÏthtac ston opo–o mporo‘me na or–soume tautÏqrona tic tuqa–ec metablhtËcqwr–c na allàxoume thn katanom† touc. Pio sugkekrimËna, an h tuqa–a metablht† X Ëqei oriste– stonq∏ro pijanÏthtac (⌦1,F1,P1) kai h tuqa–a metablht† Y Ëqei oriste– ston q∏ro pijanÏthtac (⌦2,F2,P2)mporo‘me na or–soume tic X, Y ston koinÏ deigmatikÏ q∏ro ⌦ = ⌦1 ⇥ ⌦2 wc ex†c

X(!1,!2) = X(!1), Y (!1,!2) = Y (!2).

An efodiàsoume ton ⌦ me Ëna mËtro pijanÏthtac P pou, periorismËno sto ⌦i, taut–zetai me to Pi giai 2 {1, 2}, an dhlad†

P⇥A1 ⇥ ⌦2

⇤= P1

⇥A1

⇤kai P

⇥⌦1 ⇥A2

⇤= P2

⇥A2

⇤, (6.4)

95

gia kàje Ai ⇢ ⌦i sto ped–o orismo‘ Fi tou Pi, tÏte oi X, Y , pou e–nai apÏ koino‘ orismËnec ston ⌦,Ëqoun thn –dia katanom† me tic X kai Y , ant–stoiqa. Pràgmati,

P⇥X 2 C

⇤= P

⇥{X 2 C}⇥ ⌦2

⇤= P1

⇥X 2 C

⇤

kai ant–stoiqa gia tic Y kai Y . 'Ena mËtro pijanÏthtac P pou ikanopoie– tic (6.4) onomàzetai mËtros‘zeuxhc twn P1 kai P2.

Parathr†ste Ïti en∏ oi sqËseic (6.4) pou epibàlloume sto P kajor–zoun thn katanom† kajemiàc apÏ ticX kai Y , den mac d–noun kamià plhrofor–a gia ton trÏpo pou oi X kai Y sundËontai metax‘ touc. Hteqnik† thc s‘zeuxhc sun–statai sto na epilËxoume to mËtro s‘zeuxhc P me trÏpo ∏ste h apÏ koino‘katanom† twn X, Y kàtw apÏ to P na mac odhge– se qr†sima sumperàsmata.

Ac do‘me p∏c mporo‘me na efarmÏsoume to parapànw genikÏ sqËdio mËsa apÏ d‘o sugkekrimËna para-de–gmata.

Paràdeigma 48 'Estw X tuqa–a metablht† me timËc sto R. An oi sunart†seic f, g : R ! R e–nai kaioi d‘o a‘xousec † kai oi d‘o fj–nousec, tÏte

E⇥f(X)g(X)

⇤� E

⇥f(X)

⇤E⇥g(X)

⇤

opoted†pote oi parapànw anamenÏmenec timËc e–nai kalà orismËnec.

Se autÏ to shme–o ax–zei na dokimàsete na apode–xete ton isqurismÏ tou parade–gmatoc qwr–c na qrhsi-mopoi†sete thn teqnik† thc s‘zeuxhc. Ja de–te Ïti den e–nai kajÏlou e‘kolo. Jewr†ste t∏ra mia àllhtuqa–a metablht† Y me thn –dia katanom† Ïpwc h X kai anexàrthth apÏ th X. Parathr†ste Ïti, efÏsonoi f, g Ëqoun to –dio e–doc monoton–ac, tÏte gia kàje x, y 2 R Ëqoume

�f(x)� f(y)

��g(x)� g(y)

�� 0

kai epomËnwc h tuqa–a metablht†�f(X) � f(Y )

��g(X) � g(Y )

�pa–rnei mÏno mh arnhtikËc timËc. Eidi-

kÏtera,

0 E⇥�f(X)� f(Y )

��g(X)� g(Y )

�⇤

= E⇥f(X)g(X)

⇤+ E

⇥f(Y )g(Y )

⇤� E

⇥f(X)g(Y )

⇤� E

⇥f(Y )g(X)

⇤

= E⇥f(X)g(X)

⇤+ E

⇥f(Y )g(Y )

⇤� E

⇥f(X)

⇤E⇥g(Y )

⇤� E

⇥f(Y )

⇤E⇥g(X)

⇤

= 2�E⇥f(X)g(X)

⇤� E

⇥f(X)

⇤E⇥g(X)

⇤�,

Ïpou h proteleuta–a isÏthta isq‘ei epeid† oi X,Y e–nai anexàrthtec, en∏ h teleuta–a isÏthta prok‘pteiapÏ to Ïti oi X,Y Ëqoun thn –dia katanom†.

Po‘ akrib∏c qrhsimopoi†same th s‘zeuxh sto parapànw epiqe–rhma; LËgontac ‘jewr†ste t∏ra mia àllhtuqa–a metablht† Y me thn –dia katanom† Ïpwc h X kai anexàrthth apÏ th X’ aposiwp†same p∏c mpore–kane–c na or–sei ston –dio q∏ro pijanÏthtac d‘o tuqa–ec metablhtËc me dedomËnh katanom† ∏ste autËcna e–nai anexàrthtec. AutÏ g–netai akrib∏c me thn teqnik† thc s‘zeuxhc. Me ton sumbolismÏ pouperigràyame prin, an h X e–nai orismËnh se Ënan q∏ro pijanÏthtac (⌦1,F1,P1) kai h Y e–nai orismËnhse Ënan q∏ro pijanÏthtac (⌦2,F2,P2), efodiàzoume ton q∏ro ginÏmeno ⌦1 ⇥ ⌦2 me to mËtro ginÏmenoP = P1 ⇥ P2, pou ikanopoie– thn

P⇥A1 ⇥A2

⇤= P1

⇥A1

⇤P2⇥A2

⇤,

gia kàje Ai ⇢ ⌦i sto ped–o orismo‘ Fi tou Pi. To P e–nai mËtro s‘zeuxhc afo‘ Pi

⇥⌦i

⇤= 1, epomËnwc oi

X kai Y pou e–nai orismËnec ston ⌦ Ëqoun thn katanom† twn X kai Y , ant–stoiqa. EpiplËon to P kàneitic X kai Y anexàrthtec, afo‘

P⇥X 2 C1, Y 2 C2

⇤= P

⇥{X 2 C1}⇥ {Y 2 C2}

⇤= P1

⇥X 2 C1

⇤P2⇥Y 2 C2

⇤= P

⇥X 2 C1

⇤P⇥Y 2 C2

⇤.

96

Kataskeuàsame loipÏn anexàrthtec tuqa–ec metablhtËc X, Y , orismËnec se Ënan koinÏ q∏ro pijanÏthtac,me thn –dia katanom† Ïpwc oi X kai Y ant–stoiqa. 'Eqontac plËon dei p∏c g–netai autÏ, mporo‘me qàrinaplÏthtac na sumbol–zoume tic X, Y me X kai Y ant–stoiqa.

Parathr†ste Ïti kàje mËtro s‘zeuxhc d–nei stic X, Y thn katanom† twn X,Y ant–stoiqa. EpilËgontacna efodiàsoume ton ⌦ me to mËtro ginÏmeno P kajor–same epiplËon p∏c autËc sundËontai metax‘ touc:e–nai anexàrthtec. Sth sunËqeia qrhsimopoi†same thn anexarths–a touc sthn pore–a thc apÏdeixhc.

Paràdeigma 49 Jewr†ste Ënan sunektikÏ gràfo G me s‘nolo koruf∏n V (|V | < +1) kai s‘noloakm∏n E. Gia kàje akm† tou gràfou kai anexàrthta apÏ tic àllec akmËc, apofas–zoume an ja thdiagràyoume me pijanÏthta p 2 (0, 1). 'Estw C to endeqÏmeno o gràfoc na paramËnei sunektikÏc metàtic diagrafËc. Ja de–xoume Ïti h pijanÏthta tou C e–nai fj–nousa sunàrthsh tou p.

E–nai diaisjhtikà anamenÏmeno Ïti Ïso pio pijanÏ e–nai na diagràyoume kàje akm† tÏso ligÏtero pijanÏe–nai to na parame–nei sunektikÏc o gràfoc metà tic diagrafËc. Par' Ïla autà den e–nai e‘kolo na apode–xeikane–c austhrà autÏn ton isqurismÏ qwr–c na qrhsimopoi†sei thn idËa thc s‘zeuxhc. O lÏgoc e–nai Ïti,an qrhsimopoi†sei kane–c Ëna kËrma pou Ëqei pijanÏthta na fËrei kefal† p1 kai to str–yei gia kàje akm†tou gràfou gia na apofas–sei an ja th diagràyei kai sth sunËqeia kànei to –dio me Ëna kËrma pou ËqeipijanÏthta na fËrei kefal† p2 > p1, e–nai dunatÏn ta apotelËsmata twn striyimàtwn na e–nai tËtoia ∏stesthn pr∏th per–ptwsh na katal†xei me Ënan mh sunektikÏ gràfo, en∏ sth de‘terh me Ënan sunektikÏgràfo. Ja qrhsimopoi†soume thn idËa thc suzeuxhc ∏ste na kànoume tic diagrafËc me pijanÏthta p1 kaip2 suntonismËna kai me trÏpo ∏ste oi akmËc pou diagràfoume sthn pr∏th per–ptwsh na e–nai upos‘noloeke–nwn pou diagràfoume sth de‘terh.

O gràfoc tou peiràmatoc e–nai mia tuqa–a metablht† G orismËnh ston deigmatikÏ q∏ro ⌦ = {0, 1}E metimËc sto s‘nolo twn gràfwn pou Ëqoun korufËc ta shme–a tou V kai akmËc kàpoio upos‘nolo tou E.Pràgmati, to ! = {!e}e2E 2 ⌦ antistoiqe– ston gràfo G(!) me s‘nolo akm∏n to E(!) = {e 2 E :!e = 1}. ProkeimËnou na diagràfoume kàje akm† anexàrthta me pijanÏthta p, efodiàzoume ton ⌦ me tomËtro ginÏmeno Pp, kàtw apÏ to opo–o oi tuqa–ec metabhtËc {Xe}e2E me Xe(!) = !e e–nai anexàrthteckai isÏnomec me katanom† Bernoulli(1-p).

An str–youme d‘o diaforetikà kËrmata me pijanÏthtec na fËroun kefal† p1, p2 ant–stoiqa (p1 < p2),efodiàzoume ton ⌦ ⇥ ⌦ me to mËtro s‘zeuxhc Pp1 ⇥ Pp2 kàtw apÏ to opo–o oi Xe(!,!0) = !e kaiYe(!,!0) = !0

e e–nai anexàrthtec. AutÏ to mËtro Ïmwc epitrËpei se mia akm† na diagrafe– apÏ ton G(!),Ïqi Ïmwc kai apÏ ton G(!0). Pràgmati,

Pp1 ⇥ Pp2

⇥!e = 0,!0

e = 1⇤= p1(1� p2) > 0.

Gia na exasfal–soume Ïti Ïlec oi akmËc tou G(!0) periËqontai stic akmËc tou G(!) ja kataskeuàsoumeËna mËtro s‘zeuxhc P twn Pp1 kai Pp2 ston q∏ro ginÏmeno ⌦⇥ ⌦ tËtoio ∏ste

P⇥Ye Xe gia kàje e 2 E

⇤= 1. (6.5)

Den e–nai d‘skolo na dei kane–c Ïti gia na e–nai to P mËtro s‘zeuxhc twn Pp1 kai Pp2 ja prËpei oi{Xe}e2E na e–nai anexàrthtec tuqa–ec metablhtËc me katanom† Bernoulli(1� p1) kai oi {Ye}e2E na e–naianexàrthtec tuqa–ec metablhtËc me katanom† Bernoulli(1� p2).

Mporo‘me na kataskeuàsoume Ëna mËtro P ston ⌦ ⇥ ⌦ me tic parapànw idiÏthtec wc ex†c. Arqikàefodiàzoume to {0, 1}⇥ {0, 1} me to mËtro pijanÏthtac Q gia to opo–o

Q⇥{(0, 0)}

⇤= p1, Q

⇥{(1, 0)}

⇤= p2 � p1, Q

⇥{(1, 1)}

⇤= 1� p2, kai Q

⇥{(0, 1)}

⇤= 0.

97

Sth sunËqeia or–zoume gia (!,!0) 2 ⌦⇥ ⌦

P⇥{(!,!0)}

⇤=Y

e2EQ⇥{(!e,!

0e)}

⇤.

Parathr†ste Ïti kàtw apÏ to mËtro P oi {(Xe, Ye)}e2E e–nai mia akolouj–a apÏ anexàrthtec isÏnomectuqa–ec metablhtËc me katanom† Q. EfÏson P

⇥Xe = 1

⇤= Q

⇥{(1, 0), (1, 1)}

⇤= 1 � p1 Ëqoume Ïti oi

{Xe}e2E e–nai anexàrthtec, isÏnomec tuqa–ec metablhtËc me katanom† Bernoulli(1 � p1). Ant–stoiqa,efÏson P

⇥Ye = 1

⇤= Q

⇥{(0, 1), (1, 1)}

⇤= 1 � p2, oi {Ye}e2E e–nai anexàrthtec, isÏnomec tuqa–ec meta-

blhtËc me katanom† Bernoulli(1� p2). EpomËnwc to P e–nai pràgmati mËtro s‘zeuxhc. Kàtw apÏ to P oiXe kai Ye den e–nai metax‘ touc anexàrthtec, allà gia kàje e 2 E Ëqoume

P⇥Ye Xe

⇤= Q

⇥{(0, 0), (1, 0), (1, 1)}

⇤= p1 + (p2 � p1) + (1� p2) = 1.

EpiplËon, lÏgw thc anexarths–ac twn {(Xe, Ye)}e2E , Ëqoume Ïti

P⇥Ye Xe gia kàje e 2 E

⇤= P

⇥ \

e2E{Ye Xe}

⇤=Y

e2EP⇥Ye Xe

⇤= 1,

opÏte to mËtro P pràgmati ikanopoie– thn (6.5). Ja de–xoume t∏ra Ïti an upàrqei mËtro s‘zeuxhc P twnPp1 kai Pp2 pou ikanopoie– thn (6.5), tÏte Pp1

⇥C⇤� Pp2

⇥C⇤.

An or–soume thn tuqa–a metablht†

F (!) =

(1, an o G(!) e–nai sunektikÏc

0, an o G(!) den e–nai sunektikÏc,

Ëqoume profan∏cPp

⇥C⇤= Pp

⇥{! 2 ⌦ : G(!) sunektikÏc}

⇤= Ep

⇥F⇤.

EpiplËon an !,!0 2 ⌦ kai !0e !e gia kàje e 2 E, tÏte to s‘nolo twn akm∏n tou G(!0) e–nai upos‘nolo

twn akm∏n tou G(!) kai àra F (!0) F (!). Or–zoume tËloc tic sunart†seic F1, F2 ston ⌦⇥⌦ me t‘poucF1(!,!0) = F (!) kai F2(!,!0) = F (!0). Parathr†ste Ïti, an to mËtro P ikanopoie– thn (6.5), tÏte

P⇥F1 � F2

⇤= P

⇥{(!,!0) 2 ⌦⇥ ⌦ : F (!) � F (!0)}

⇤� P

⇥!0e !e, gia kàje e 2 E

⇤= 1. (6.6)

'Eqoume t∏raPp1

⇥C⇤= Ep1

⇥F⇤= E

⇥F1⇤� E

⇥F2⇤= Ep2

⇥F⇤= Pp2

⇥C⇤,

Ïpou h de‘terh kai h proteleuta–a isÏthta prok‘ptoun apÏ to gegonÏc Ïti to P e–nai mËtro s‘zeuxhc,en∏ h tr–th isÏthta apÏ thn (6.6).

Ja kle–soume aut† thn paràgrafo me mia parat†rhsh. Sthn apÏdeixh pou d∏same gia ton isqurismÏtou Parade–gmatoc 49, qrhsimopoi†same sumbolismÏ sumbatÏ me autÏn thc eisagwg†c thc paragràfou,kataskeuàzontac Ëna katàllhlo mËtro s‘zeuxhc ston q∏ro ginÏmeno {0, 1}E ⇥ {0, 1}E . DiabàzontacÏmwc prosektikà thn apÏdeixh diapist∏noume Ïti e–nai arketÏ na kataskeuàsoume tuqa–ec metablhtËc{Xe, Ye}e2E se Ënan koinÏ q∏ro pijanÏthtac (⌦,F ,P) Ëtsi ∏ste: a) oi {Xe}e2E na e–nai anexàrthtectuqa–ec metablhtËc me katanom† Bernoulli(1�p1), b) oi {Ye}e2E na e–nai anexàrthtec tuqa–ec metablhtËcme katanom† Bernoulli(1� p2) kai g) P

⇥Ye Xe gia kàje e 2 E

⇤= 1.

AutÏ mporo‘me na to pet‘qoume me ton ex†c aplo‘stero kai endeqomËnwc pio didaktikÏ trÏpo. Jewro‘-me Ënan q∏ro pijanÏthtac (⌦,F ,P) ston opo–o mporo‘me na or–soume mia akolouj–a apÏ anexàrthtec,

98

isÏnomec tuqa–ec metablhtËc {Ue}e2E me omoiÏmorfh katanom† sto (0,1). Gia kàje e 2 E or–zoume ticXe, Ye me th bo†jeia thc Ue wc ex†c

Xe = {Ue > p1} kai Ye = {Ue > p2}.

LÏgw thc anexarths–ac twn {Ue}e2E oi {(Xe, Ye)}e2E e–nai anexàrthtec. EpiplËon

P⇥Xe = 1

⇤= P

⇥Ue > 1� p1

⇤= 1� p1 kai P

⇥Ye = 1

⇤= P

⇥Ue > 1� p2

⇤= 1� p2,

àra pràgmati oi (a) kai (b) parapànw ikanopoio‘ntai. TËloc h (g) e–nai profan†c apÏ thn kataskeu† twnXe, Ye afo‘ Ye = 1 , Ue > p2 ) Xe = 1.

6.4 Asumptwtik† katanom†

H paràgrafoc aut† e–nai afierwmËnh sthn apÏdeixh tou parakàtw Jewr†matoc.

Je∏rhma 28 'Estw {Xn}n2N0 mia mh upobibàsimh, gnhs–wc epanalhptik† kai aperiodik† alus–da stonq∏ro katastàsewn X. An ⇡n(x) = P

⇥Xn = x

⇤gia x 2 X, e–nai h katanom† thc alus–dac metà apÏ n 2 N0

b†mata kai ⇡ e–nai h anallo–wth katanom† thc alus–dac, tÏte

limn!1

⇡n = ⇡.

EidikÏtera,limn!1

p(n)(x, y) = ⇡(y), gia kàje x 2 X.

ApÏdeixh: H apÏdeixh bas–zetai sthn teqnik† thc s‘zeuxhc. 'Opwc e–dame sthn paràgrafo thc s‘zeu-xhc, mporo‘me na kataskeuàsoume ston –dio q∏ro pijanÏthtac thn alus–da {Xn}n2N0 kai mia àllh ane-xàrthth alus–da {Yn}n2N me arqik† katanom† ⇡ kai tic –diec pijanÏthtec metàbashc Ïpwc h {Xn}n2N0 .Or–zoume ton qrÏno

T = inf{k � 0 : Xk = Yk}

thc pr∏thc sunànthshc twn d‘o anexàrthtwn alus–dwn. To upÏloipo thc apÏdeixhc mpore– na qwriste–se mia seirà apÏ b†mata.

B†ma 1: Ja de–xoume Ïti P⇥T < +1

⇤= 1.

Ston q∏ro katastàsewn X⇥X jewro‘me th stoqastik† diadikas–a {Wn}n2N0 me Wn = (Xn, Yn). E–naie‘kolo na dei kane–c Ïti aut† e–nai mia markobian† alus–da me arqik† katanom† ⇡0 Ïpou

⇡0�(x, u)

�= P

⇥W0 = (x, u)

⇤= P

⇥X0 = x, Y0 = u

⇤= ⇡0(x)⇡(u)

kai pijanÏthtec metàbashc

p�(x, u), (y, v)

�= P

⇥Wn+1 = (y, v) |Wn = (x, u)

⇤= P

⇥Xn+1 = y, Yn+1 = v |Xn = x, Yn = u

⇤

= P⇥Xn+1 = y |Xn = x

⇤P⇥Yn+1 = v |Yn = u

⇤= p(x, y)p(u, v).

EpiplËon h {Wn}n e–nai mh upobibàsimh. Pràgmati, efÏson h {Xn} e–nai aperiodik†, apÏ to L†mma 9 giakàje x, y, u, v 2 X upàrqei Ëna n0 tËtoio ∏ste

n � n0 ) p(n)(x, y) > 0 kai p(n)(u, v) > 0.

99

'Etsi, an n � n0, Ëqoume

p(n)�(x, u), (y, v)

�= P

⇥Wn = (y, v) |W0 = (x, u)

⇤= P

⇥Xn = y, Yn = v |X0 = x, Y0 = u

⇤

= P⇥Xn = y |X0 = x

⇤P⇥Yn = v |Y0 = u

⇤= p(n)(x, y)p(n)(u, v) > 0,

epomËnwc opoiad†pote katàstash (y, v) 2 X ⇥ X e–nai prosbàsimh apÏ opoiad†pote àllh katàstash(x, u) 2 X⇥ X.

Ja de–xoume tËloc Ïti h ⇡ : X ⇥ X ! [0, 1] me ⇡(y, v) = ⇡(y)⇡(v) e–nai anallo–wth katanom† gia th{Wn}n. Pràgmati, e–nai fanerÏ Ïti ⇡(y, v) � 0 gia kàje (y, v) 2 X⇥ X, en∏

X

y,v2X⇡(y, v) =

X

y,v2X⇡(y)⇡(v) =

X

y2X⇡(y)

X

v2X⇡(v) = 1.

TËloc,X

x,u2X⇡(x, u)p

�(x, u), (y, v)

�=

X

x,u2X⇡(x)⇡(u)p(x, y)p(u, v)

=X

x2X⇡(x)p(x, y)

X

u2X⇡(u)p(u, v) = ⇡(y)⇡(v) = ⇡(y, v).

EfÏson h {Wn}n e–nai mia mh upobibàsimh alus–da pou Ëqei anallo–wth katanom† ⇡, e–nai kai gnhs–wcepanalhptik†. O qrÏnoc T pou or–same parapànw e–nai o qrÏnoc pr∏thc àfixhc thc {Wn}n sth diag∏niotou X⇥X, dhlad† sto s‘nolo � = {(w,w) : w 2 X}. ApÏ thn epanalhptikÏthta thc alus–dac prok‘pteio isqurismÏc tou b†matoc.

B†ma 2: Or–zoume th diadikas–a {Zn}n2N0 wc ex†c:

Zn(!) =

(Xn(!), an n T (!)

Yn(!), an n > T (!).

Ja de–xoume Ïti h {Zn}n e–nai mia markobian† alus–da ston X me arqik† katanom† ⇡0 kai pijanÏthtecmetàbashc p(·, ·), Ëqei dhlad† thn –dia katanom† me th {Xn}n.

AutÏc o isqurismÏc e–nai diaisjhtikà fanerÏc. H {Zn} parakolouje– th {Xn} mËqri ton qrÏno T katàton opo–o h {Xn}n sunantà gia pr∏th forà thn {Yn}n kai sth sunËqeia parakolouje– thn {Yn}n. Epeid†Ïmwc, apÏ thn isqur† markobian† idiÏthta, tÏso h {XT+n}n2N0 Ïso kai h {YT+n}n2N0 e–nai markobianËcalus–dec pou xekino‘n apÏ to shme–o sunànths†c touc, Ëqoun pijanÏthtec metàbashc p(·, ·) kai e–naianexàrthtec apÏ to pareljÏn tou qrÏnou T , perimËnoume Ïti ja e–nai ad‘nato na xeqwr–sei kane–c thstatistik† sumperiforà twn {Xn}n kai {Zn}n.

To Ïti h Z0 Ëqei katanom† ⇡0 e–nai fanerÏ, afo‘ T � 0 kai àra P⇥Z0 = X0

⇤= 1. Ja de–xoume akÏma

Ïti an z0, z1, . . . , zn, zn+1 2 X, tÏte

P⇥Zn+1 = zn+1, Zn = zn, . . . , Z0 = z0

⇤= p(zn, zn+1) P

⇥Zn = zn, . . . , Z0 = z0

⇤. (6.7)

ApÏ thn (6.7) Ëpetai Ïti h {Zn}n2N0 e–nai markobian†, me pijanÏthtec metàbashc {p(x, y)}x,y2X. ApÏ thdiamËrish ⌦ =

Sn

k=0{T = k} [ {T > n} Ëqoume

P⇥Zn+1 = zn+1, Zn = zn, . . . , Z0 = z0

⇤=

nX

k=0

P⇥Zn+1 = zn+1, Zn = zn, . . . , Z0 = z0, T = k

⇤

+ P⇥Zn+1 = zn+1, Zn = zn, . . . , Z0 = z0, T > n

⇤. (6.8)

100

Gia k 2 {0, 1, . . . , n}, Ëqoume

{Zn+1=zn+1, . . . , Z0=z0, T =k} =k\

i=0

{Xi=zi}k�1\

i=0

{Yi 6=zi}n+1\

i=k

{Yi=zi}.

ApÏ thn anexarths–a twn alus–dwn {Xn}n2N0 , {Yn}n2N0 kai th markobian† idiÏthta thc {Yn}n2N0 pa–r-noume Ïti

P⇥Zn+1=zn+1, . . . , Z0=z0, T =k

⇤= P

⇥ k\

i=0

{Xi=zi}⇤P⇥ k�1\

i=0

{Yi 6=zi}n+1\

i=k

{Yi=zi}⇤

= P⇥ k\

i=0

{Xi=zi}⇤P⇥ k�1\

i=0

{Yi 6=zi}n\

i=k

{Yi=zi}⇤p(zn, zn+1)

= P⇥ k\

i=0

{Xi=zi}k�1\

i=0

{Yi 6=zi}n\

i=k

{Yi=zi}⇤p(zn, zn+1)

= P⇥Zn=zn, . . . , Z0=z0, T =k

⇤p(zn, zn+1). (6.9)

Me ant–stoiqo trÏpo, Ëqoume Ïti

{Zn+1=zn+1, . . . , Z0=z0, T > n} =n+1\

i=0

{Xi=zi}n\

i=0

{Yi 6=zi},

en∏ apÏ thn anexarths–a twn d‘o alus–dwn kai th markobian† idiÏthta thc {Xn}n2N0 pa–rnoume Ïti

P⇥Zn+1=zn+1, . . . , Z0=z0, T > n

⇤= P

⇥Zn=zn, . . . , Z0=z0, T > n

⇤p(zn, zn+1). (6.10)

Antikajist∏ntac tic (6.9) kai (6.10) sthn (6.8) pa–rnoume thn (6.7).

B†ma 3: Ja de–xoume Ïti, gia kàje x 2 X, Ëqoume

|⇡n(x)� ⇡(x)| P⇥T > n

⇤. (6.11)

EfÏson oi {Xn}n kai {Zn}n Ëqoun thn –dia katanom†, Ëqoume

⇡n(x) = P⇥Xn = x

⇤= P

⇥Zn = x

⇤, gia kàje x 2 X, n 2 N0.

Epeid† h arqik† katanom† ⇡ thc {Yn}n e–nai anallo–wth katanom† Ëqoume

⇡(x) = P⇥Yn = x

⇤, gia kàje x 2 X, n 2 N0.

Sunduàzontac tic d‘o parapànw sqËseic Ëqoume Ïti, gia kàje x 2 X kai n 2 N0,��⇡n(x)� ⇡(x)| = | P

⇥Zn = x

⇤� P

⇥Yn = x

⇤ ��

=�� P

⇥Zn = x, T n

⇤+ P

⇥Zn = x, T > n

⇤� P

⇥Yn = x, T n

⇤� P

⇥Yn = x, T > n

⇤ ��

=�� P

⇥Xn = x, T > n

⇤� P

⇥Yn = x, T > n

⇤ ��

P⇥T > n

⇤.

H akolouj–a endeqomËnwn An = {T > n} e–nai fj–nousa, dhlad† An+1 ⇢ An, gia kàje n 2 N0. 'EqoumeloipÏn

limn!1

P⇥T > n

⇤= P

⇥\

n

{T > n}⇤= P

⇥T = 1

⇤= 0

101

apÏ to apotËlesma tou b†matoc 1. Sunep∏c, gia kàje x 2 X Ëqoume ⇡n(x) ! ⇡(x) kai àra ⇡n ! ⇡.

TËloc, gia na apode–xoume Ïti p(n)(x, y) ! ⇡(y) arke– na xekin†soume thn alus–da {Xn} apÏ thn ka-tàstash x, na pàroume dhlad† P

⇥X0 = x

⇤= 1. TÏte p(n)(x, y) = P

⇥Xn = y

⇤! ⇡(y), mia kai o

isqurismÏc tou jewr†matoc isq‘ei gia opoiad†pote arqik† katanom† ⇡0.

Parathr†ste p∏c qrhsimopoi†same th sunj†kh thc aperiodikÏthtac sthn apÏdeixh sto b†ma 1. Qwr–cthn aperiodikÏthta thc {Xn} den e–nai dunatÏn na sumperànoume Ïti h {Wn}n e–nai mh upobibàsimh. Angia paràdeigma jewr†soume mia alus–da pou kine–tai stic korufËc enÏc tetrag∏nou A,B,C,D kai sekàje b†ma metaba–nei me pijanÏthta 1/2 se m–a apÏ tic d‘o geitonikËc thc korufËc, tÏte h katàstash(A,A) den e–nai prosbàsimh gia thn {Wn}n apÏ thn katàstash (A,B). Pràgmati,

P⇥Wn = (A,A) |W0 = (A,B)

⇤= p(n)(A,A)p(n)(B,A) = 0,

afo‘ o pr∏toc Ïroc tou ginomËnou sto dex– mËloc e–nai 0 an n perittÏc, en∏ o de‘teroc Ïroc e–nai 0an n àrtioc. EpomËnwc, qwr–c thn aperiodikÏthta thc {Xn} den mporo‘me na sumperànoume th sunj†khP⇥T < +1

⇤= 1 pou mac d–nei to zhto‘meno me th bo†jeia thc ekt–mhshc tou b†matoc 3.

6.5 To ergodikÏ je∏rhma

Se aut† thn paràgrafo ja apode–xoume to ergodikÏ je∏rhma gia markobianËc alus–dec. To ergodikÏ je-∏rhma mac d–nei plhrofor–ec gia thn asumptwtik† sumperiforà tou qroniko‘ mËsou Ïrou sunarthsiak∏nthc alus–dac kai me aut† thn Ënnoia e–nai mia gen–keush tou klasiko‘ nÏmou twn megàlwn arijm∏n sthnper–ptwsh metablht∏n pou den e–nai anexàrthtec. Sthn per–ptwsh mh arnhtik∏n tuqa–wn metablht∏npou ja qreiasto‘me parakàtw o isqurÏc nÏmoc twn megàlwn arijm∏n diatup∏netai wc ex†c.

Je∏rhma 29 'Estw {Zn}n2N mia akolouj–a apÏ anexàrthtec, isÏnomec, mh arnhtikËc tuqa–ec meta-blhtËc me E

⇥Zi

⇤= µ 1. TÏte

P⇥ 1

n

nX

k=1

Zk ! µ⇤= 1.

H apÏdeixh auto‘ tou jewr†matoc mpore– na breje– sthn [8] kai sthn [9]. ProsËxte Ïti gia kàje n 2 N oqronikÏc mËsoc thc akolouj–ac 1

n

Pn

k=1 Zk e–nai mia tuqa–a metablht†. O nÏmoc twn megàlwn arijm∏n màcd–nei thn plhrofor–a Ïti h katanom† aut†c thc tuqa–ac metablht†c sugkentr∏netai asumptwtikà g‘rw apÏto µ. H anexarths–a twn {Zn}N2N pa–zei apofasistikÏ rÏlo se autÏ. Pràgmati, sthn akra–a per–ptwshpou pàrei kane–c Ïlec tic metablhtËc –sec me mia tuqa–a metablht† Z, o mËsoc Ïroc oswnd†pote apÏ autËce–nai h pàli h Z kai fusikà h katanom† tou den sugkentr∏netai g‘rw apÏ Ënan arijmÏ. Mporo‘me nakatalàboume ton mhqanismÏ me ton opo–on upeisËrqetai h anexarths–a sthn per–ptwsh µ < 1 wc ex†c.Gia Ëna tupikÏ ! 2 ⌦, kàpoiec apÏ tic tuqa–ec metablhtËc pa–rnoun timËc megal‘terec apÏ th mËsh tim†touc µ kai kàpoiec mikrÏterec. Epeid† oi metablhtËc e–nai anexàrthtec, oi proc ta pànw apokl–seic apo thmËsh tim† se megàlo bajmÏ allhloanairo‘ntai me tic proc ta kàtw apokl–seic, tÏso pou oi ajroistikËcapokl–seic n prosjetËwn e–nai pol‘ mikrÏterec apÏ to n me to opo–o diairo‘me kai sto Ïrio n ! 1exafan–zontai.

An t∏ra {Xn}n2N0 e–nai mia mh upobibàsimh markobian† alus–da se Ënan q∏ro katastàsewn X kaif : X ! R, mpore– kane–c na melet†sei ton qronikÏ mËso Ïro twn pragmatik∏n tuqa–wn metablht∏nZn = f(Xn). AutËc bËbaia den e–nai anexàrthtec. 'Eqoume Ïmwc dei Ïti h exàrthsh thc katanom†c miacmh upobibàsimhc markobian†c alus–dac apÏ tic prohgo‘menec timËc thc exasjene– proÏntoc tou qrÏnou.To ergodikÏ Je∏rhma mac lËei Ïti kai se aut† thn per–ptwsh oi ajroistikËc apokl–seic n prosjetËwn

102

e–nai pol‘ mikrÏterec apÏ to n kai oi qroniko– mËsoi Ïroi sugkl–noun kaj∏c n ! 1. Sthn aplo‘sterhmorf† tou pou ja do‘me pr∏ta, h sunàrthsh f e–nai h de–ktria miac katàstashc x 2 X kai to ergodikÏje∏rhma mac d–nei plhrofor–a gia to asumptwtikÏ posostÏ tou qrÏnou pou pernà h alus–da sth x.

Gia x 2 X or–zoume Vn(x) to pl†joc twn episkËyewn sth x miac alus–dac {Xn}n2N0 prin th qronik†stigm† n,

Vn(x) =n�1X

k=0

{Xk = x}.

Je∏rhma 30 (ergodikÏ Je∏rhma) 'Estw {Xn}n2N0 mia mh upobibàsimh markobian† alus–da se ËnandiakritÏ q∏ro katastàsewn X. Gia opoiad†pote arqik† katanom† thc alus–dac, Ëqoume

Ph Vn(x)

n! 1

Ex

⇥T+x

⇤ , gia kàje x 2 Xi= 1,

Ïpou T+x = inf{k > 0 : Xk = x} e–nai o qrÏnoc pr∏thc epistrof†c thc alus–dac sthn katàstash x.

ApÏdeixh: Ja apode–xoume pr∏ta Ïti gia kàje x 2 X

Ph Vn(x)

n! 1

Ex

⇥T+x

⇤i= 1. (6.12)

Ax–zei na parathr†soume Ïti to an upàrqei to Ïrio Vn(x)/n kai to poio e–nai autÏ exartàtai mÏno apÏthn telik† sumperiforà thc alus–dac kai Ïqi apÏ ta osad†pote pr∏ta b†matà thc. Pràgmati, an M 2 Nkai jewr†soume th diadikas–a {Yn}n2N0 me Yn = XM+n, Ëqoume

n�1X

k=0

{Xk = x}�n�1X

k=0

{Yk = x} =M�1X

k=0

�{Xk = x}� {Xn+k = x}

�

kai epomËnwc�� 1n

n�1X

k=0

{Xk = x}� 1

n

n�1X

k=0

{Yk = x}�� M

n! 0, kaj∏c n ! 1.

Màlista, akÏma ki an h M = M(!) e–nai Ënac tuqa–oc qrÏnoc me P⇥M < +1

⇤= 1, mporo‘me na

qrhsimopoi†soume autÏ to epiqe–rhma gia kàje ! 2 {! 2 ⌦ : M(!) < +1}.

An h x e–nai epanalhptik† katàstash kai Tx = inf{k � 0 : Xk = x}, tÏte gia opoiad†pote arqik†katanom† thc alus–dac Ëqoume P

⇥Tx < +1

⇤= 1. EpiplËon, apÏ thn isqur† markobian† idiÏthta, h

diadikas–a {Yn}n2N0 me Yn = XTx+n e–nai mia alus–da pou xekinà apÏ to x kai Ëqei tic –diec pijanÏthtecmetàbashc Ïpwc h {Xn}n. S‘mfwna me thn prohgo‘menh parat†rhs† mac, gia na de–xoume thn (6.12)sthn per–ptwsh pou h x e–nai epanalhptik†, arke– na de–xoume Ïti

Px

h Vn(x)

n! 1

Ex

⇥T+x

⇤i= 1. (6.13)

Or–zoume S0 = 0 kai epagwgikà gia kàje n 2 N ton qrÏno thc n-st†c epanÏdou sthn katàstash x wcex†c

Sn(!) = inf{k > Sn�1(!) : Xk = x}.

ApÏ thn isqur† markobian† idiÏthta kai thn epanalhptikÏthta thc x, Ëqoume Px

⇥Sn < +1 gia kàje n 2

N⇤= 1. EpiplËon, Ïpwc e–dame sto PÏrisma 2, oi qrÏnoi {Sn � Sn�1}n2N, pou mesolabo‘n anàmesa

103

se diadoqikËc episkËyeic sthn katàstash x e–nai anexàrthtec, isÏnomec tuqa–ec metablhtËc me koin†katanom† aut†n thc S1�S0 = T+

x . ApÏ ton nÏmo twn megàlwn arijm∏n (Je∏rhma 29) Ëqoume epomËnwc

Px

hlimn!1

Sn

n= Ex

⇥T+x

⇤i= 1.

ApÏ thn epanalhptikÏthta thc katàstashc x Ëqoume akÏma

Px

⇥Vn(x) ! 1

⇤= 1.

Afo‘ ta parapànw endeqÏmena sumba–noun me Px-pijanÏthta 1 kai h tom† touc ja sumba–nei me Px-pijanÏthta 1, epomËnwc

Px

hlimn!1

SVn(x)

Vn(x)= Ex

⇥T+x

⇤i= 1. (6.14)

Parathr†ste Ïmwc t∏ra Ïti, gia mia alus–da pou xekinà apÏ to x, o qrÏnoc SVn(x) = inf{k � n : Xk = x}e–nai o qrÏnoc thc pr∏thc ep–skeyhc sthn x metà th qronik† stigm† n gia kàje n 2 N, epomËnwc

Px

hSVn(x)�1 < n SVn(x), gia kàje n 2 N

i= 1

kai àra h (6.13) prok‘ptei apÏ thn (6.14).

An h katàstash x e–nai parodik†, h apÏdeixh thc (6.12) e–nai akÏma pio e‘kolh, afo‘ tÏte Ex

⇥T+x

⇤= +1,

en∏ P⇥limn!1 Vn(x) < +1

⇤= 1 kai àra P

⇥limn!1 Vn(x)/n = 0

⇤= 1.

O isqurismÏc tou jewr†matoc prok‘ptei t∏ra apÏ thn (6.12) epeid† o X Ëqei to pol‘ arijm†simo pl†jocapÏ stoiqe–a. Pràgmati, h (6.12) shma–nei Ïti gia kàje x 2 X Ëqoume P

⇥Ux

⇤= 1, Ïpou

Ux =�! 2 ⌦ :

Vn(x)

n! 1

Ex

⇥T+x

⇤ .

EfÏson kajËna apÏ ta Ux Ëqei pijanÏthta 1, to –dio ja sumba–nei kai gia thn arijm†simh tom† touc\x2XUx, afo‘

Ph� \

x2XUx

�ci= P

h [

x2XU c

x

iX

x2XP⇥U c

x

⇤=X

x2X0 = 0.

Parat†rhsh: An h alus–da {Xn} e–nai ektÏc apÏ mh upobibàsimh kai gnhs–wc epanalhptik†, ja Ëqeimonadik† anallo–wth katanom† ⇡, h opo–a ja ikanopoie– Ïpwc e–dame sto prohgo‘meno kefàlaio thn

⇡(x) =1

Ex

⇥T+x

⇤ , gia kàje x 2 X.

Se aut† thn per–ptwsh to Je∏rhma 30 màc lËei Ïti se bàjoc qrÏnou to posostÏ tou qrÏnou pou xode‘eih alus–da se kàje katàstash x 2 X e–nai to bàroc pou d–nei sth x h anallo–wth katanom† ⇡:

Ph Vn(x)

n! ⇡(x), gia kàje x 2 X

i= 1. (6.15)

Sthn per–ptwsh aut† mporo‘me na diatup∏soume to ergodikÏ Je∏rhma kai me mia diaforetik† morf† poupollËc forËc e–nai pio qr†simh gia upologismo‘c.

104

Je∏rhma 31 (ergodikÏ Je∏rhma II) 'Estw {Xn}n2N0 mia mh upobibàsimh, gnhs–wc epanalhptik† mar-kobian† alus–da se Ënan diakritÏ q∏ro katastàsewn X. An f : X ! R e–nai mia fragmËnh sunàrthsh,tÏte gia opoiad†pote arqik† katanom† thc alus–dac Ëqoume

Ph 1

n

n�1X

k=0

f(Xk) ! E⇡⇥f⇤i

= 1,

Ïpou E⇡⇥f⇤=P

x2X f(x)⇡(x) e–nai h mËsh tim† thc tuqa–ac metablht†c f(X), an h tuqa–a metablht† Xakolouje– thn anallo–wth katanom† thc alus–dac ⇡.

ApÏdeixh: Epeid† kàje qronik† stigm† k 2 N0 h alus–da br–sketai se akrib∏c m–a apÏ tic katastàseictou X Ëqoume X

x2X{Xk = x} = 1.

Mporo‘me t∏ra na gràyoume ton qronikÏ mËso twn tim∏n thc f katà m†koc tou monopatio‘ thc alus–dacwc

1

n

n�1X

k=0

f(Xk) =1

n

n�1X

k=0

f(Xk)X

x2X{Xk = x}

=1

n

X

x2X

n�1X

k=0

f(Xk) {Xk = x}

=1

n

X

x2X

n�1X

k=0

f(x) {Xk = x}

=X

x2Xf(x)

1

n

n�1X

k=0

{Xk = x}

=X

x2Xf(x)

Vn(x)

n.

Sthn per–ptwsh pou o X e–nai peperasmËnoc, o isqurismÏc prok‘ptei kateuje–an apÏ to Je∏rhma 30.Sthn per–ptwsh pou o X Ëqei àpeiro allà arijm†simo pl†joc apÏ katastàseic, jewr†ste M > 0 tËtoio∏ste |f(x)| M , gia kàje x 2 X. EfÏson h seirà

Px2X ⇡(x) sugkl–nei (sto 1), gia kàje ✏ > 0

mporo‘me na bro‘me Ëna peperasmËno s‘nolo apÏ katastàseic A ⇢ X tËtoio ∏steP

x/2A ⇡(x) < ✏

2M .'Eqoume

���1

n

n�1X

k=0

f(Xk)� E⇡⇥f⇤��� =

���X

x2Xf(x)

�Vn(x)

n� ⇡(x)

����

MX

x2X

���Vn(x)

n� ⇡(x)

���

= MX

x2A

���Vn(x)

n� ⇡(x)

���+MX

x/2A

���Vn(x)

n� ⇡(x)

���. (6.16)

Gia touc prosjetËouc tou teleuta–ou Ïrou ja qrhsimopoi†soume thn algebrik† tautÏthta |u| = u+2u�,pou isq‘ei gia kàje u 2 R, Ïpou u� e–nai to arnhtikÏ mËroc tou u kai d–netai apÏ thn u� = max{0,�u}.

105

Pa–rnoume Ëtsi

X

x/2A

���Vn(x)

n� ⇡(x)

��� =X

x/2A

⇣Vn(x)

n� ⇡(x)

⌘+ 2

X

x/2A

⇣Vn(x)

n� ⇡(x)

⌘�

=X

x2A

⇣⇡(x)� Vn(x)

n

⌘+ 2

X

x/2A

⇣Vn(x)

n� ⇡(x)

⌘�

X

x2A

⇣⇡(x)� Vn(x)

n

⌘+ 2

X

x/2A

⇡(x)

X

x2A

⇣⇡(x)� Vn(x)

n

⌘+

✏

M,

Ïpou sto de‘tero b†ma qrhsimopoi†same Ïti

X

x2X

Vn(x)

n= 1 =

X

x2X⇡(x),

en∏ sto tr–to Ïti, an u, v > 0, tÏte (u� v)� v. Me antikatàstash thc parapànw sqËshc sthn (6.16)Ëqoume

���1

n

n�1X

k=0

f(Xk)� E⇡⇥f⇤��� 2M

X

x2A

���Vn(x)

n� ⇡(x)

���+ ✏.

EfÏson to s‘nolo A e–nai peperasmËno, gia ta ! 2 ⌦ gia ta opo–a Vn(x)/n ! ⇡(x) Ëqoume

lim supn!1

���1

n

n�1X

k=0

f(Xk)� E⇡⇥f⇤��� ✏

kai àra apÏ to Je∏rhma 30 Ëqoume Ïti gia kàje ✏ > 0

Phlim supn!1

���1

n

n�1X

k=0

f(Xk)� E⇡⇥f⇤��� ✏

i= 1.

An pàroume t∏ra ✏ = 1/N , h akolouj–a endeqomËnwn

BN = {! 2 ⌦ : lim supn!1

���1

n

n�1X

k=0

f(Xk)� E⇡⇥f⇤���

1

N}

e–nai fj–nousa (BN+1 ⇢ BN gia kàje N 2 N) kai P⇥BN

⇤= 1 gia kàje N 2 N. EpomËnwc,

Ph 1

n

n�1X

k=0

f(Xk) ! E⇡⇥f⇤i

= P⇥\

N

BN

⇤= lim

N!1P⇥BN

⇤= 1,

pou e–nai kai o isqurismÏc tou jewr†matoc.

106

6.6 Ask†seic

'Askhsh 95 Apode–xte to Je∏rhma 27.

'Askhsh 96 Upolog–ste th sundiak‘mansh twn tuqa–wn metablht∏n Xe kai Ye tou Parade–gmatoc 49.

'Askhsh 97 TËsseric upologistËc e–nai sundedemËnoi anà d‘o. Kàje tËtoia s‘ndesh e–nai leitourgik†me pijanÏthta p 2 [0, 1] anexàrthta apÏ tic àllec sundËseic. Upolog–ste thn pijanÏthta f(p) na upàrqeileitourgikÏ kanàli pou na epitrËpei thn epikoinwn–a d‘o dedomËnwn upologist∏n (endeqomËnwc kai mËswtwn àllwn d‘o upologist∏n) kai melet†ste th sunàrthsh f : p ! f(p) wc proc th monoton–a thc. SthsunËqeia de–xte Ïti h f e–nai a‘xousa me thn teqnik† thc s‘zeuxhc kai parathr†ste pÏso pio e‘koloc kaigenike‘simoc se pio pol‘plokec sundesmolog–ec e–nai o de‘teroc trÏpoc.

'Askhsh 98 R–qnoume epanalambanÏmena Ëna t–mio zàri kai prosjËtoume ta apotelËsmata twn ende–-xewn. An Sn e–nai to àjroisma twn n pr∏twn zari∏n mac, upolog–ste to Ïrio

limn!1

P⇥Sn e–nai pollaplàsio tou 7

⇤.

'Askhsh 99 Sthn 'Askhsh 5, se bàjoc qrÏnou, poio posostÏ tou qrÏnou tou xode‘ei to Ëntomo stosalÏni;

'Askhsh 100 Sthn 'Askhsh 81, se bàjoc qrÏnou, poio posostÏ twn hmer∏n to mpakàliko den ËqeimpiskÏta thn ∏ra pou kle–nei;

'Askhsh 101 Sto parakàtw sq†ma fa–nontai oi katastàseic kai oi pijanÏthtec metàbashc miac alu-s–dac {Xn}n2N0 .

1

2

13

23 3

2/31/3

1/32/3

41/3

52/3

1/212

61/2

14

7

8

121

1/4

1/2

a) Taxinom†ste tic katastàseic se klàseic epikoinwn–ac.

b) Bre–te Ïlec tic anallo–wtec katanomËc thc alus–dac.

g) An X0 = 5, upolog–ste thn pijanÏthta h alus–da na katal†xei se kajem–a apÏ tic kleistËc klàseic.

d) Upolog–ste to Ïrio limn!1

P⇥Xn = j |X0 = i

⇤gia ta parakàtw ze‘gh (i, j): (5,5), (8,8) kai (5,8)

e) 'Estw X0 = 5. Me anàlush pr∏tou b†matoc † me opoiond†pote àllon trÏpo, upolog–ste thn pija-nÏthta h alus–da na egkatale–yei thn klàsh apÏ thn opo–a xekinà Ëpeita apÏ Ënan àrtio arijmÏ bhmàtwn.

'Askhsh 102 An h Xn e–nai mia markobian† alus–da s' Ëna q∏ro X, tÏte h Yn+1 = (Xn, Xn+1) e–naimia markobian† alus–da ston X⇥X. Poiec e–nai oi pijanÏthtec metàbashc thc Y ; An h X Ëqei monadik†anallo–wth katanom† ⇡, de–xte Ïti h y Ëqei monadik† anallo–wth katanom† thn ⇡⇤(j, k) = ⇡(j)p(j, k).

'Askhsh 103 Me th bo†jeia thc prohgo‘menhc àskhshc de–xte Ïti an h alus–da {Xn}n2N e–nai mhupobibàsimh kai kine–tai s' Ënan peperasmËno q∏ro katastàsewn X kai f : X⇥X ! R, tÏte me pijanÏthta1 Ëqoume

limn!1

f(X0, X1) + f(X1, X2) + · · ·+ f(Xn�1, Xn)

n=

X

(x,y)2X⇥Xf(x, y)⇡(x)p(x, y).

gia opoiad†pote katanom† thc X0.

107

'Askhsh 104 Sthn 'Askhsh 79, se bàjoc qrÏnou, poio posostÏ twn metabàsewn g–nontai metax‘ twnkatastàsewn 3 kai 4; Poio e–nai to Ïrio

limn!1

P⇥Xn = 4|Xn+1 = 3

⇤;

'Askhsh 105 Mia markobian† alus–da sto N mpore– na metabe– apÏ Ënan fusikÏ n � 2 e–te ston n+1e–te ston n� 1. 'Otan br–sketai sto 1 mpore– e–te na pàei sto 2 e–te na parame–nei sto 1. Oi pijanÏthtecmetàbashc ikanopoio‘n thn

p(n, n+ 1) =n

n+ 2p(n+ 1, n) > 0 gia kàje n 2 N.

Me mia pr∏th matià fa–netai Ïti oi parapànw sunj†kec den kajor–zoun pl†rwc tic pijanÏthtec metàbashcallà h p(1, 1) = x e–nai mia ele‘jerh paràmetroc. Oi àllec pijanÏthtec metàbashc upolog–zontai me thbo†jeia thc x wc ex†c:

p(1, 2) = 1� x, p(2, 1) =1 + 2

1p(1, 2) = 3(1� x), p(2, 3) = 1� p(2, 1) = 3x� 2 k.o.k.

a) Ekfràste thn p(n, n + 1) wc sunàrthsh thc x kai de–xte Ïti p(n, n + 1) 2 (0, 1] gia kàje n 2 N ankai mÏno an x = 4 ln(2)� 2.

An katafËrate na de–xete ton prohgo‘meno isqurismÏ, s–goura ja kànate arket† douleià. Ja do‘me t∏raËnan pol‘ aplÏ trÏpo gia na upolog–soume thn p(1, 1).

b) Bre–te mia katanom† ⇡ sto N pou ikanopoie– tic sunj†kec akribo‘c isorrop–ac me tic {p(m,n)}m,n2N.

g) Qrhsimopoi∏ntac to ergodikÏ je∏rhma de–xte Ïti to Ïrio limn!1(�1)X1+···+(�1)Xn

nupàrqei me pija-

nÏthta 1 kai upolog–ste to.

d) Sto prohgo‘meno àjroisma upàrqoun pollËc allhloanairËseic anàmesa se diadoqiko‘c Ïrouc thcalus–dac. Prospaj†ste na katalàbete poioi Ïroi dhmiourgo‘n th mh mhdenik† suneisforà sto Ïrio kaisumperànete Ïti p(1, 1) = 4 ln(2)� 2.

'Askhsh 106 'Enac pa–kthc tou mpàsket propone–tai sta tr–ponta. 'Eqete parathr†sei Ïti h pija-nÏthta na eustoq†sei se Ëna sout e–nai –sh me

• 1/2 an Ëqei eustoq†sei kai sta d‘o prohgo‘mena sout pou Ëqei epiqeir†sei

• 1/3 an Ëqei eustoq†sei se Ëna apÏ ta d‘o prohgo‘mena sout pou Ëqei epiqeir†sei

• 1/4 an Ëqei astoq†sei kai sta d‘o prohgo‘mena sout pou Ëqei epiqeir†sei.

Gia n 2 N or–zoume Xn = 1 an to n-ostÏ sout tou pa–kth e–nai e‘stoqo kai 0 diaforetikà.

a. E–nai h akolouj–a {Xn} markobian†; Dikaiolog†ste thn apànths† sac.b. E–nai h alus–da Yn = (Xn, Xn+1) mia markobian† alus–da ston q∏ro katastàsewnX = {0, 1}⇥ {0, 1} = {(0, 0), (0, 1), (1, 0), (1, 1)}; Poiec e–nai oi pijanÏthtec metàbashc;g. Poia e–nai h anallo–wth katanom† aut†c thc alus–dac;d. Se bàjoc qrÏnou ti posostÏ apÏ ta sout tou e–nai e‘stoqa;

'Askhsh 107 EpilËgoume tuqa–a d‘o n-y†fiouc arijmo‘c kai touc prosjËtoume. AnAn e–nai to pl†joctwn krato‘menwn pou metafËrame katà thn prÏsjesh, ti sumba–nei sto Ïrio limn

Ann;

'Askhsh 108 Sto parakàtw sq†ma fa–nontai oi katastàseic kai oi pijanÏthtec metàbashc miac mar-kobian†c alus–dac {Xn}n.

108

! "

#

$

%

&

' (

1/4

3/43/4

1/4

1/4 1/4

1/41/41/2

1/4 1/4

1 1/4

1/4

1/4

1/4

3/4

1/43/4 1/4

a) Bre–te tic klàseic epikoinwn–ac thc alus–dac kai qarakthr–ste tic wc proc thn epanalhptikÏthta.b) An X0 = 3, poioc e–nai o anamenÏmenoc qrÏnoc exÏdou thc alus–dac apÏ thn klàsh pou periËqei thnkatàstash 3 kai poia h pijanÏthta na katal†xei se kajemià apÏ tic àllec klàseic;g) Upolog–ste to Ïrio limn P

⇥Xn = 8 |X0 = 8

⇤.

d) An X0 = 8, T1 = inf{k > 0 : Xk = 8} kai T2 = inf{k > T1 : Xk = 8}, dhlad† T1 kai T2 e–nai oiqrÏnoi pr∏thc kai de‘terhc epanÏdou sto 8 ant–stoiqa, upolog–ste tic E

⇥T1⇤kai E

⇥T2⇤.

e) 'Estw X0 = 3. An kerd–zete 1 eur∏ kàje forà pou h alus–da br–sketai se katàstash me àrtio de–kth,ti mpore–te na pe–te gia to mËso kËrdoc sac anà k–nhsh se bàjoc qrÏnou; Poiec timËc mpore– na pàrei;Me poia pijanÏthta;

6.7 Arijmhtikà peiràmata

'Askhsh 109 Sthn àskhsh aut† g–netai prosomo–wsh thc alus–dac thc 'Askhshc 83 gia p = 1/2.Upenjum–zetai Ïti aut† h alus–da ston q∏ro katastàsewn N [ {0} e–nai mh upobibàsimh, gnhs–wc epa-nalhptik†, me anallo–wth katanom† ⇡ pou d–netai apÏ thn

⇡(k) =1

2k+1, k = 0, 1, 2, 3, . . .

Katebàste kai trËxte ton k∏dika ergodic.py. O k∏dikac autÏc prosomoi∏nei ta pr∏ta N = 106 b†matathc alus–dac kai epistrËfei ton ergodikÏ mËso

X1 +X2 + · · ·+XN

N.

ApÏ to ergodikÏ Je∏rhma h parapànw posÏthta sugkl–nei kaj∏c N ! 1 me pijanÏthta 1 sto

1X

k=0

k⇡(k),

opÏte o k∏dikac upolog–zei arijmhtikà thn parapànw posÏthta me th mËjodo Markov Chain MonteCarlo (MCMC).

a) Upolog–ste analutikà to parapànw àjroisma kai epibebai∏ste Ïti o k∏dikac prosfËrei mia kal†ekt–mhs† tou.

b) Allàxte ton k∏dika ∏ste na upolog–zei to parapànw àjroisma 50 forËc kai bre–te th diasporà twnapotelesmàtwn.

109

g) An jËlame na perior–soume th diasporà twn apotelesmàtwn sto misÏ, pÏso megàlo ja Ëprepe napàroume to N ; Apant†ste jewrhtikà kai epibebai∏ste to arijmhtikà.

d) Allàxte ton k∏dika ∏ste na upolog–zei me th mËjodo MCMC to àjroisma

1X

k=0

cos(k + cos(k))

2k.

'Askhsh 110 S' Ëna paiqn–di Ëqete to dika–wma na r–xete Ëna zàri Ïsec forËc jËlete kai na eispràxeteto àjroisma twn zari∏n sac an den fËrete kanËnan àso. An se kàpoia zarià fËrete àso tÏte to paiqn–ditelei∏nei qwr–c na kerd–sete t–pota. JËloume na megistopoi†soume to anamenÏmeno kËrdoc apÏ to pai-qn–di. PerimËnoume Ïti h bËltisth strathgik† ja †tan na stamat†soume mÏlic to àjroisma twn zari∏nmac xeperàsei kàpoio kat∏fli k, to opo–o jËloume na prosdior–soume arijmhtikà.

O k∏dikac opt stop.py prosomoi∏nei gia k = 0, 1, . . . , 40 N = 1.000 part–dec tou paiqnidio‘ me kat∏flidiakop†c k kai upolog–zei to mËso kËrdoc me kàje strathgik†. Sthn ËxodÏ tou epistrËfei th strath-gik† me thn opo–a shmei∏jhke to megal‘tero kËrdoc kai Ëna diàgramma me to upologizÏmeno kËrdoc wcsunàrthsh tou k.

a) TrËxte ton k∏dika kai de–te to diàgramma merikËc forËc. Parathr†ste Ïti h megàlh diasporà sthnekt–mhsh tou kËrdouc den mac epitrËpei na prosdior–soume th bËltisth strathgik†.

Ja do‘me t∏ra p∏c mporo‘me na antimetwp–soume to prÏblhma thc diasporàc. H mËjodoc Monte Carlomac prosfËrei Ënan arijmhtikÏ trÏpo upologismo‘ miac posÏthtac x, an mporo‘me na fantasto‘me th xwc thn anamenÏmenh tim† kàpoiac tuqa–ac metablht†c X pou mporo‘me na prosomoi∏soume. H ektim†triaMonte Carlo thc x

xN =X1 + · · ·+XN

N

sugkl–nei sth x apÏ to nÏmo twn megàlwn arijm∏n kaj∏c N ! 1, en∏ h diasporà thc gia peperasmËnoN e–nai �

2

N, Ïpou �2 e–nai h diasporà thc X. Sto paiqn–di mac, to kËrdoc se m–a part–da e–nai mia tuqa–a

metablht† me megàlh diasporà �2. Pa–rnei e–te thn tim† mhdËn, an fËroume àso prin xeperàsoume tokat∏fli k, e–te kàpoia apÏ tic timËc k+2, k+3, . . . , k+6. 'Etsi h diasporà thc ektim†triàc mac e–nai kiaut† sqetikà megàlh.

b) 'Enac trÏpoc na elatt∏soume th diasporà thc ektim†triac e–nai na aux†soume to N kai epomËnwc tonupologistikÏ qrÏno. Dekaplasiàste to N kai epanalàbete to prohgo‘meno er∏thma. 'Eluse aut† hallag† entel∏c to prÏblhma pou e–qame;

'Enac àlloc trÏpoc e–nai na fantasto‘me to x wc th mËsh tim† miac tuqa–ac metablht†c Y me mikrÏterhdiasporà apÏ th X. Upàrqoun pollo– trÏpoi na g–nei autÏ kai anafËrontai sth bibliograf–a wc teqnikËcelàttwshc diasporàc (variance reduction). Ja dokimàsoume sto paiqn–di mac thn teqnik† thc deigmato-lhy–ac katà shmantikÏthta (importance sampling).

An r–qname Ëna zàri qwr–c àso, to kËrdoc mac se m–a part–da me kat∏fli diakop†c k ja Ëpairne mÏnokàpoia apÏ tic timËc k + 2, k + 3, . . . , k + 6 kai ja e–qe pol‘ mikrÏterh diasporà. Ja prospaj†soumeloipÏn na ekfràsoume to mËso kËrdoc E

⇥K⇤sto arqikÏ mac paiqn–di me kat∏fli diakop†c k mËsw tou

kËrdouc pou ja e–qe aut† h strathgik†, an to zàri mac den e–qe àso. Parathro‘me arqikà Ïti

E⇥K⇤=X

m

X

(x0,x1,...,xm)2Jk

xmP⇥X1 = x1, . . . , Xm = xm

⇤

=X

m

X

(x0,x1,...,xm)2Jk

xm⇣16

⌘m

.

110

Ïpou

Jk = {(x0, x1, . . . , xm) : x0 = 0, xi+1 2 {xi + 2, . . . , xi + 6}, i = 0, 1, . . . ,m� 1, xm�1 k, xm > k}

e–nai eke–nec oi troqiËc pou xeperno‘n to k se m zariËc, qwr–c endiàmesa na Ëqei Ërjei àsoc. An t∏ra{Yn}n e–nai h alus–da twn skor mac Ïtan r–qnoume Ëna zàri qwr–c àso, Ëqoume Ïti

P⇥Yn+1 = ` |Yn = j

⇤=

1

5, ` 2 {j + 2, . . . , j + 6}.

Mporo‘me epomËnwc na gràyoume

E⇥K⇤=X

m

X

(x0,x1,...,xm)2Jk

xm⇣56

⌘m

P⇥Y1 = x1, . . . , Ym = xm

⇤= E

hK 0� 5

6

�Mi, (6.17)

Ïpou K 0 e–nai to kËrdoc mac kai M e–nai o arijmÏc twn zari∏n pou r–xame prokeimËnou na xeperàsoumeto kat∏fli k me to peiragmËno zàri.

g) Qrhsimopoi†ste thn (6.17) gia na upolog–sete me Monte Carlo poia e–nai h bËltisth strathgik†diakop†c kai poio e–nai to anamenÏmeno kËrdoc mac apÏ to paiqn–di me aut†n th strathgik†.

111