MARKOV CHAINS. 7. Convergence to equilibrium. Long-run proportions. Part IB Michaelmas 2009 YMS. Proof. (a) state j we have π (i) P ) = π

Transcription

1 Part IB Mchaelmas 2009 YMS MARKOV CHAINS E-mal: 7 Covergece to equlbrum Log-ru proportos Covergece to equlbrum for rreducble, postve recurret, aperodc chas ad proof by couplg Log-ru proporto of tme spet a gve state Covergece to equlbrum meas that, as the tme progresses, the Markov cha forgets about ts tal dstrbuto λ I partcular, f λ = δ (), the Drac delta cocetrated at, the cha forgets about tal state Clearly, ths s related to propertes of the -step matrx P as Cosder frst the case of a fte cha Theorem 7 Suppose that a fte m m trasto matrx P coverges, each etry, to a lmtg matrx P = (π j ): lm p() j = π j,, j I (7) The (a) every row π () of P s a equlbrum dstrbuto π () P = π () or π j = l π l p lj (b) If P s rreducble the all rows π () cocde: π () = = π (m) = π I ths case, lm P(X = j) = π j j I ad the tal dstrbuto λ Proof (a) state j we have ( π () P ) = π j l p lj l I = lm l p() l p lj = lm l p () l p lj = lm p (+) j = π j = ( π ()) j (72) (b) If P s rreducble the all rows π () of P cocde as there s a uque equlbrum dstrbuto Also, lm P(X = j) = lm λ p () j = λ lm p () j = π j (73) For a coutable cha, our argumet Eq (72) requres a justfcato of exchagg the order of the lmt ad summato I ll omt ths argumet: the reader ca fd t the recommeded lterature We see from Theorem 7 that the equlbrum dstrbuto of a cha ca be detfed from the lmt of matrces P as More precsely, f we kow that P coverges to a matrx P whose rows are equal to each other the these rows gve the equlbrum dstrbuto π We see therefore that covergece P Π where Π has a structure factor π π So whe does P Π? A smple couterexample: P = ( ) 0 P =, eve, ( 0 ) 0 P =, odd 0 More geerally, cosder a m m matrx P = s a crucal ( ) 0 Here 0 (74) I ths case the equlbrum dstrbuto s uque: π = (/2, /2) but there s o covergece P Π, as P s perodc (of perod 2)

2 correspodg to Fg 7 m 2 Fgure 7 3 m 2 Fgure 73 3 The P 2 wll correspod to 2 m Fgure 72 Smlarly, for hgher powers, wth the mth power P m = I: 3 The pcture wll the be repeated mod m Aga, the equlbrum dstrbuto s uque: π = (/m,/m), but the covergece P Π fals Defto 7 Trassto matrx P s called aperodc f I p () > 0 for all large eough (75) If addto, P s rreducble the,, j I p () j > 0 for all large eough (76) Theorem 72 Assume P s rreducble, aperodc ad postve recurret The, as, P Π The etres of the lmtg matrx Π are costat alog colums I other words the rows of Π are repettos of the same vector π whch s the (uque) equlbrum dstrbuto for P Hece, the rreducble aperodc ad postve recurret Markov cha forgets ts tal dstrbuto: λ ad j I, lm P(X = j) = π j Proof: a sketch (No-examable but useful may stuatos) Cosder two Markov chas (X () ), whch s ( δ (), P ), ad (X π ), whch s 3 4

3 (π, P) The p () j = P (X () = j), π j = P(X π = j) To evaluate the dfferece betwee these probabltes, we wll detfy ther commo part, by couplg the two Markov chas, e rug them together Oe way s to ru both chas depedetly It meas that we cosder the Markov cha (Y ) o I I, wth states (k, l) where k, l I, wth the trasto probabltes ad wth the tal dstrbuto p Y (k,l)(u,v) = p ku p lv, k, l, u, v I, (77) P(Y 0 = (k, l)) = (k = )π l, k, l I However, a better way for us s to ru the cha (W ) where the trasto probabltes are { p W (k,l)(u,v) = p ku p lv, f k l, k, l, u, v I, (78) p ku (u = v), f k = l, wth the same tal dstrbuto P(W 0 = (k, l)) = (k = )π l, k, l I (79) Ideed, Eq (78) determes a trasto probablty matrx o I I: all etres p W (k,l)(u,v) 0 ad the sum alog a row equals oe I fact, p ku p lv, f k l p W (k,l)(u,v) = u v = p ku, f k = l u,v I u Further, the partal summato gves the orgal trastoal probabltes P p W (k,l)(u,v) = p ku, p W (k,l)(u,v) = p lv v I Pctorally, the two compoets of the cha (W ) behave dvdually lke (X () ) ad (X π); together they evolve depedetly (e as (Y )) utl the (radom) tme T whe they cocde u I T = f [ : X () = Xπ ], 5 after whch they stay together Therefore, Wrtg ad P W ( X () p () j π j = P ( W X () = j) P W (X π = j) = j) = P W ( X () = j, T ) + P W ( X () = j, T > ) (70) P W (X π = j) = P W (X π = j, T ) + P W (X π = j, T > ), (7) we see that the frst summads cacel each other: P ( W X () = j, T ) = P W (X π = j, T ), { } as the evets X () = j, T ad {X π = j, T } cocde Hece ad p () j π j = P ( W X () = j, T > ) P W (X π = j, T > ) p () j π j P W (T > ) = P Y (T > ) (72) The last boud s called the couplg equalty Thus, t suffces to check that P W (T > ) 0, e P(T < ) = Ths s establshed by usg the fact that the orgal matrx P s rreducble ad aperodc (I omt the detals) I the case of a fte rreducble aperodc cha t s possble to establsh that the rate of covergece of p () j to π j s geometrc I fact, ths case m ad ρ (0, ) such that Theorem 73 p (m) j ρ states, j (73) If P s fte rreducble ad aperodc the states, j p () j π j ( ρ) /m, (74) 6

4 where m ad ρ are as (73) Proof (No-examable but useful may stuatos) Repeat the scheme of the proof of Theorem 72: we have to assess P Y (T > ) But the fte case, we ca wrte e P W (k,l)(t m) u I p (m) ku p(m) lu ρ p (m) lu = ρ, u I P W (k,l)(t > m) ( ρ) k, l I The, by the strog Markov property [ ) P W (T > ) P (T W > m P m] W (T > m) [/m] ad the asserto of Theorem 73 follows Examples 7 Cosder a m m stochastc matrx whose rows are cyclc shfts of a gve stochastc vector (p,,p m ) where p,, p m > 0 ad p + + p m = : p p 2 p m p m p 2 p 3 p m p P = p m p p m Sce all states commucate drectly, [ ths matrx s rreducble ] ad aperodc; moreover, the value m 0 = m : p () j > 0, j I = The equlbrum dstrbuto s uque: π = (/m,,/m) By Theorems 7 ad 72, P Π geometrcally fast: π j ( ρ) p () j where ρ = m [ p,,p m ] (0, ) 7 72 (Card shufflg) The problem of shufflg a pack of cards s mportat ot oly gamblg but a umber of other applcato See Example Sheet 2 Remark 7 For a traset or ull recurret rreducble aperodc cha, matrx P coverges to a zero matrx: lm P = O We wll ot gve here the formal proof of ths asserto (For a traset case the proof s based o the fact that the seres p () < ) Defto 72 Cosder the umber of vsts to state before tme : V () = (X k = ) (77) The lmt (f t exsts) V () lm s called the log-ru proporto of the tme spet state Theorem 74 state I: where r = k=0 (78) ( ) V () P lm = r =, (79) { π, f s postve recurret, 0, f s ull recurret or traset (720) Proof Frst, suppose that state s traset The, as we kow, the total umber V of vsts to s fte wth probablty See Eqs (58), 8

5 (58) Hece, V / 0 as wth probablty As 0 V () V, we deduce that V ()/ 0 as wth probablty Now let be recurret The the tmes T (), T (2), betwee successve returs to state are fte wth P -probablty By Theorem 65, they are IID radom varables, wth mea value m equal to /π the postve recurret case ad to the ull recurret case Obvously, but see Fg 74 So, we ca wrte: ( V () T () T () + + T (V()), T () + + T (V() ), ) + + T (V() ) V () ( V () T () ) + + T (V()) By Theorem 66, o a evet of P -probablty, the lmt lm m holds: P ( l= T (l) m, as 9 ) (72) T (l) = l= = (722) state ( ) ( ) ( ) T T 2 T 3 Fgure 74 H ( ) V _ ( ) T ( ) _ V ( ) _ () V X T ( ) V ( ) tme tme Next, as s recurret, sequece (V ()) creases deftely, aga o a evet of P -probablty : ) P (V () ր, as = (723) The we ca put (722) a summato up to V (), stead of ad, correspodgly, dvde by the factor V (): V () lm T (l) = m V () l= Ths relato holds o the tersecto of the two aforemetoed evets of probablty, whch obvously has aga P -probablty O the same evet, lm V () V () l= T (l) = m I other words, Eqs (722) ad (723) together yeld that P V () V T (l) () m ad T (l) m, as = V () V () l= 0 l= (724)

6 But the, owg to (72), stll o the same tersecto of two evets of P -probablty, the rato /V () teds to m, e the verse rato V ()/ teds to r = /m Ths gves (79), (720) ad completes the proof of Theorem 74 Remark 73 A careful aalyss of the proof of Theorem 74 shows that f P s rreducble ad postve recurret, the we ca clam that (79) the probablty dstrbuto P ca be replaced by P j, or, fact, by the dstrbuto P geerated by a arbtrary tal dstrbuto λ Ths s possble because sums T () + + T () stll behave asymptotcally as f the RVs T (l) were IID (I realty, the dstrbuto of the frst RV, T () = T = H, wll be dfferet ad deped o the choce of the tal state) Theorem 75 Let P be a fte rreducble trasto matrx The for ay tal dstrbuto λ ad a bouded fucto f o I: ( ) V (f, ) P lm = π(f) =, (725) where π(f) = I π f() (726) Proof The proof of Theorem 75 s a re-femet of that of Theorem 74 More precsely, (725) s equvalet to ( ) P lm V (f, ) π(f) = 0 = I other words, we have to check that o a evet of P-probablty, V (f, ) π(f) 0, as (727) Wrtg V (f, ) = V ()f() ad π(f) = π f(), we ca trasform I I ad boud the left-had sde (727) as follows V (f, ) π(f) = ( ) V () π f() V () I I π f() We kow that, I, o a evet of P -probablty, V ()/ π Remark 7 allows us to clam covergece V ()/ π o a evet of P j - probablty (that s, regardless of the choce of the tal state), or, eve stroger, o a evet of P-probablty, where P s the dstrbuto of the (λ, P) Markov cha wth ay tal dstrbuto λ The (725) follows, whch completes the proof of Theorem 75 Example 73 (Markov Chas, Part IIA, 2002, A40M) Wrte a essay o the log-tme behavour of dscrete tme Markov chas o a fte state space Your essay should clude dscusso of the covergece of probabltes as well as almost-sure behavour You should also expla what happes whe the cha s ot rreducble Soluto The state space splts to ope classes O,, O j ad closed classes C j+,, C j+l If l = (a uque closed class), t s rreducble Startg from a ope class, say O, we ed up closed class C k wth probablty h k These probabltes satsfy j+l h k = p r h k r r= Here, p r s the probablty that we ext class O to class O r or C r, ad for r = j +,, j + l: h k r = δ r,k The cha has a uque equlbrum dstrbuto π (r) cocetrated o C r, r = j+,, j+l (hece, a uque equlbrum dstrbuto whe l = ) Ay equlbrum dstrbuto s a mxture of the equlbrum dstrbutos π (r) Startg C r, we have, for ay fucto f o C r : t=0 f(x t ) π (r) f() almost surely C r 2

7 Moreover, the aperodc case (where gcd { : p aa () > 0} = for some a C r ), 0 C r : P(X = X 0 = 0 ) π r, ad the covergece s wth a geometrc speed 8 Detaled balace ad reversblty Tme reversal, detaled balace, reversblty; radom walk o a graph Let (X 0, X, ) be a Markov cha ad fx N What ca we say about the tme reversal of (X ), e the famly (X N, = 0,,, N) = (X N, X N,,X 0 )? Theorem 8 Let (X ) be a (π, P) Markov cha where π = (π ) s a equlbrum dstrbuto for P wth π > 0 I The: (a) N, the tme reversal (X N, X N,, X 0 ) s a (π, P) Markov cha where P = ( p j ) has p j = π j p j (8) π (b) If P s rreducble the so s P Proof (a) Frst, observe that P s a stochastc matrx, that s, p j 0 ad p j = π j p j = π = π π Next, π s P-varat π p j = Now pull the factor π through the product j j π j p j = π j p j = π j P (X N = N,, X 0 = 0 ) = P (X 0 = 0,,X N = N ) = π 0 p 0 p N N = p 0 π p N N = p 0 p 2 π 2 = p 0 p N N π N = π N p N N p 0 3 4

8 We see that (X N ) s a (π, P) Markov cha (b) If P s rreducble the ay par of states, j s coected, that s a path = 0,,, = j wth 0 < p 0 p = (/π 0 )π 0 p 0 p = (/π 0 ) p 0 π p = = (/π 0 ) p 0 p π So, p 0 p > 0, ad j, are coected P The case where cha (X N ) has the same dstrbuto as (X ) s of a partcular terest Theorem 82 are equvalet: () ad states 0,, : Let (X ) be a Markov cha The followg propertes P(X 0 = 0,, X = ) = P(X 0 =, X = 0 ) (82) () (X ) s equlbrum, e (X ) (π, P) where π s a equlbrum dstrbuto for P, ad So, λ = (λp), e λp = λ Hece, the cha s equlbrum λ = π Next,, j () () Wrte P(X 0 =, X = j) = π p j = P(X 0 = j, X = ) = π j p j P(X 0 = 0,,X = ) = π 0 p 0 p ad use Eqs (83) to pull π through the product π 0 p 0 p = p 0 π p = = p 0 p π = π p p 0 = P(X 0 =,,X = 0 ) Defto 8 A Markov cha (X ) satsfyg (82) s called reversble Eqs (83) are called detaled balace equatos (DBEs) So, the asserto of Theorem 82 reads; a Markov cha s reversble f ad oly f t s equlbrum, ad the DBEs are satsfed The DBEs are a powerful tool for detfcato of a ED π p j = π j p j states, j I (83) Theorem 83 If λ ad P satsfy the DBEs Proof () () Take =, P(X 0 =, X = j) = P(X 0 = j, X = ), ad sum over j P(X 0 =, X = j) = P(X 0 = ) = λ, j P(X 0 = j, X = ) = P(X = ) = (λp) j 5 λ p j = λ j p j,, j I, the λ s a ED for P, that s λp = λ Proof Sum over j: λ p j = λ, j λ j p j = (λp) The two expressos are equal, hece the result 6 j

9 So, for a gve matrx P, f the DBEs ca be solved (that s, a probablty dstrbuto that satsfes them ca be foud), the soluto wll gve a ED Furthermore, the correspodg Markov cha wll be reversble A terestg ad mportat class of Markov chas s formed by radom walks o graphs We have see examples of such chas: a brth-death process (a RW o Z or ts subset), a RW o a plae square lattce Z 2 ad, more geerally, a RW o a d-dmesoal cubc lattce Z d A feature of these examples s that a waderg partcle ca jump to ay of ts eghbourg stes; a symmetrc case, the probablty of each jump s the same Ths dea ca be exteded to a geeral graph, wth drected or o-drected lks (edges) Here, we focus o o-drected graphs; a graph s uderstood as a collecto G of vertces some of whch are joed by o-drected edges, or lks, possbly several No-drected meas here that the edges ca be traversed both drectos; sometmes t s coveet to thk that each edge s formed by a par of opposte arrows Fgure 8 A graph s called coected f ay two dstct vertces are coected wth a path formed by edges The valecy v of a vertex s defed as the umber of edges at The coectedess v j s the umber of edges jog vertces ad j The RW o the graph has the followg trasto matrx P = (p j ) { / v j v, f ad j are coected, p j = (84) 0, otherwse The matrx P s rreducble f ad oly f the graph s coected The vector 7 v = (v ) satsfes the DBEs That s, vertces, j v p j = v j = v j p j, (85) ad hece s P-varat We obta the followg straghtforward result Theorem 84 The RW o a graph, wth trasto matrx P of the form (84), could be of all three types: traset (vz, a symmetrc earesteghbour RW o Z d wth d 3), ull recurret (a symmetrc earesteghbour RW o Z 2 or Z ) or postve recurret It s postve recurret f ad oly f the total valece / v <, whch case π j = v j v s a equlbrum dstrbuto Furthermore, the cha wth equlbrum dstrbuto π s reversble A smple but popular example of a graph s a l-ste segmet of a oedmesoal lattce: here the valecy of every vertex equals 2, except for the edpots where the valece s See Fg 82 a) l = 8 = l 2 a) b) Fgure 82 A terestg class s formed by graphs wth a costat valecy: v v; aga the smplest case s v = 2, where l vertces are placed o a crcle (or o a perfect polygo or ay closed path) See Fg 82 b) A popular example 8

10 of a graph wth a costat valecy s a fully coected graph wth a gve umber of vertces, say {,, m}: here the valecy equals m, ad the graph has m(m )/2 (o-drected) edges total See Fg 83 m = 5 m = 6 a) b) Fgure 83 Aother mportat example s a regular cube d dmesos, wth 2 d vertces Here the valecy equals d, ad the graph has d2 d (stll odrected) edges jog egbourg vertces See Fg 84 d = 2 d = 3 Fgure 84 d = 4 Popular examples of fte graphs of costat valecy are lattces ad trees I the case of a geeral fte graph of costat valecy v = v vertx, the sum v equals v G where G s the umber of vertces The probabltes p j = p j = v j /v, eghbourg par, j That s, the trasto matrx P = (p j ) s Hermta: P = P T Furthermore, the equlbrum dstrbuto π = (π ) s uform: π = / G I Lear Algebra courses, t s asserted that a (complex) Hermta matrx has a orthoormal bass of ege-vectors, ad ts ege-values are all real Ths hady property s ce to reta wheever possble For a Markov cha, eve whe P s orgally o-hermta, t ca be coverted to a Hermta matrx, by chagg the scalar product We wll explore further ths aveue Sectos 2 4 Example , B0M) (Markov Chas, Part IIA, 2002, A0M ad Part IIA, () We are gve a fte set of arports Assume that betwee ay two arports, ad j, there are a j = a j flghts each drecto o every day A cofused traveller takes oe flght per day, choosg at radom from all avalable flghts Startg from, how may days o average wll pass utl the traveller returs aga to? Be careful to allow for the case where may be o flghts at all betwee two gve arports () Cosder the fte tree T wth root R, where for all m 0, all vertces at dstace 2 m from R have degree 3, ad where all other vertces (except R) have degree 2 Show that the radom walk o T s recurret Soluto () Let X 0 = be the startg arport, X the destato of the th flght ad I deote the set of arports reachable from The (X ) s a rreducble Markov cha o I, so the expected retur tme to, s gve by (/π ), where π s the uque equlbrum dstrbuto We wll show that /π = / a jk a k j,k I k I I fact, p jk = a jk a jl l I ad ( ( a jl )p jk = a kl )p kj l I l I 9 20

11 So the vector v = (v j ) wth v j = l I a jl s detaled balace wth P Hece π j = k I a jk / k,l I a kl () Cosder the dstace X from the root R at tme The (X ) 0 s a brth-death Markov cha wth trasto q = p = /2, f 2 m, q = /3, p = 2/3, f = 2 m By a stadard argumet for h = P (ht 0) ad The codto so h 0 =, h = p h + + q h,, p u + = q u, u = h h, u + = q p u = γ u, γ = q q p p, u + + u = h 0 h, h = A(γ γ ) γ = forces A = 0 ad hece h = for all Here, γ 2 m = 2 m, γ = ad the walk s recurret Assume, for defteess, that P s rreducble, ad π > 0 I The aswer comes out after we defe the trasto matrx P RV = (p RV j ) by or π p RV j = π j p j,, j I (86) p RV j = π j π p j,, j I (87) Eqs (86), (87) deed determe a trasto matrx, as, j I, p RV j 0, ad j I p RV j = π j p j = π = π π Next, π gves a ED for P RV : j I, π p RV j = π j p j = π j I I The, repeatg the argumet from the proof of Theorem 8, we obta that N, the tme reversal (X N, 0 N) s a Markov cha equlbrum, wth trasto matrx P RV ad the same ED π Symbolcally, j I (X RV ) ( π, P RV) Markovcha, (88) where (X RV ) = (X N ) stads for the tme reversal of T It s structve to remember that P RV was prove to be a stochastc matrx because π s a ED for P whle the proof that π s a ED for P RV used oly the fact that P s stochastc The DBEs are a coveet tool to fd a equlbrum dstrbuto: f a measure λ 0 s detaled balace wth P ad has λ <, the π j = λ j / λ s a equlbrum dstrbuto Example 82 Suppose π = (π ) forms a ED for trasto matrx P = (p j ), wth πp = π, but the DBE s (83) are ot satsfed What s the tme reversal of cha (X ) equlbrum? 2 Example 83 The detaled balace equatos have a useful geometrc meag Suppose that the state space I = {,,s} Matrx P geerates a lear trasformato R s R s, where vector x = s take to Px Assumg P rreducble, let π be the ED, wth π > 0, =,,s Cosder a tlted scalar product, π R s, where x,y π = x x s s x y π (89) = 22

12 The detaled balace equatos (83) mea that P s self-adjot (or Hermta) relatve to scalar product, π that s, I fact, x, Py π =,j x, Py π = Px,y π, x,y R s (80) x p j y j π =,j x p j y j π j = Px,y π The coverse s also true: Eq (80) mples (83), as we ca take as x ad y the vectors δ ad δ j wth the oly o-zero etres at postos ad j, respectvely,, j =,,s Ths observato yelds a beeft, as Hermta matrces have all egevalues real, ad ther ege-vectors are mutually orthogoal (relatve to the scalar product questo, ths stace,, π ) We wll use ths Secto 2 Remark 8 The cocept of reversblty ad tme reversal wll be partcularly helpful a cotuous-tme settg of Part II Appled Probablty It s ow tme to gve a bref summary of essetal results establshed about varous equatos emergg the aalyss of Markov chas We have see two sets of equatos: (I) for httg probabltes h A ad mea httg tmes k A ad (II) for equlbrum dstrbutos π = (π ) ad expected tmes γ k spet state before returg to k Although they are a sese smlar, there are also dffereces betwee them whch t s mportat to remember (I) For h j = P (ht j) the equatos are where h j j =, hj = l I Here, h j s always a soluto p l h j l = (h j P T ), j, h j = (h j, I), wth hj j = P T =, as (P T ) = l 23 p l = I (I2) For k j = E (tme to ht j) the equatos are k j j = 0, kj = + p l k j l = + (k j P T ), j where l I, l j k j = (k j, I), wth kj j = 0 Here, takg that 0 = 0, k j = ( δ j) s always a soluto whe the cha s rreducble These equatos are produced by codtog o the frst jump The vectors h j ad k j are labelled by the termal states whle ther etres h j ad k j dcate the tal states The soluto we look for s detfed as a mmal o-egatve soluto satsfyg the ormalsato costrats h j j = ad k j j = 0 (II) For the equatos are or γ k = E k (tme spet before returg to k) γ k k =, γk = l γ k l p l, k, γ k = γ k P, whe k s recurret Here, the codtog s o the last jump, ad vectors γ k are labelled by startg states The detfcato of the soluto s by the codtos γ k 0 ad γ k k = (II2) Smlarly, for a equlbrum dstrbuto (or more geerally, a varat measure) π = πp The detfcato here s through the codto π 0 ad (II3) A soluto to the detaled balace equatos π p j = π j p j, π = always produces a varat measure If addto, π =, t gves a equlbrum dstrbuto As the detaled balace equatos are usually easy to solve (whe they have a soluto), t s a powerful tool whch s always worth tryg whe you eed to fd a equlbrum dstrbuto 24