Radom Walks ad Browia Motio Tel Aviv Uiversity Sprig 20 Lecture date: Mar 2, 20 Lecture 4 Istructor: Ro Peled Scribe: Lira Rotem This lecture deals primarily with recurrece for geeral radom walks. We preset several criteria for a radom walk to be recurret, ad prove Polya s theorem o recurrece ad trasiece for the simple radom walk o Z d Recurrece for geeral radom walks Remember the followig result from the previous lecture: Every -dimesioal radom walk S satisfies exactly oe of the followig almost surely: (i) S = 0 for every. (ii) lims =. (iii) lims =. (iv) limsups = ad limifs =. We would like to cotrast this with the followig defiitio: Defiitio A radom walk takig values i R d is called poit-recurret if P(S = 0 ifiitely ofte) =. (remember this probability is either 0 or, by the Hewitt-Savage zero-oe law). We also defie the set of possible values for the radom walk as the set of all x R d such that P(S = x) > 0 for some. Exercise If a radom walk is poit-recurret, the for every possible value x. P(S = x ifiitely ofte) = Remark 2 If the radom walk is ot discrete (havig a atomic distributio for each step) the these defiitios are ot very useful. Istead we say that the radom walk is eighborhood-recurret if for some (ad the for ay, as ca be proved) ǫ > 0 P( S < ǫ ifiitely ofte) =. 4-
Similary, possible values are chaged to those x for which for every ǫ > 0 there exists a such that P( S x < ǫ) > 0. Propositio 3 A radom walk i R d is poit-reucrret if ad oly if oe of the followig holds: (i) P(, S = 0) =. (ii) P(S = 0 ifiitely ofte) = (this is the defiitio of poit-recurrece). (iii) P(S = 0) =. Proof The equivalece betwee (i) ad (ii) is clear. For the equivalece betwee (i) ad (iii), ote that by the strog markov property, the umber of visits to 0 is distributed geometrically Geo(p), where p is the probability of o retur to 0. Therefore P(S = 0) = E(Geo(p)) = p, ad ow the equivalece is obvious. Theorem 4 (Pólya) A simple radom walk i Z d is recurret if ad oly if d = or d = 2. Proof We cosider each dimesio separately: d = : We already saw this result i dimesio, but we ow give a differet proof. Notice that by Stirlig s formula ( ) 2 P(S 2 = 0) = 2 2. π Therefore P(S 2 = 0) = ad we are doe. d = 2: By rotatig Z 2 i 45 we get that each step is like movig oe step i each of two idepedet, oe dimesioal, simple radom walks. Hece ad agai P(S 2 = 0) =. P(S 2 = 0) ( π ) 2 = π, 4-2
d = 3: We calculate explicitly: P(S 2 = 0) = 6 2 j,k 0 j+k = 2 2 ( 2 (2)! (j! k! ( j k)!) 2 ) j,k 0 j+k ( ) 3! 2 j! k! ( j k)! (here j is the umber of steps i directio (,0,0), ad k is the umber of steps i directio (0,,0) ). Sice 3! j! k! ( j k)! =, it follows that j,k 0 j+k P(S 2 = 0) 2 2 ( 2 ) max j,k 0 j+k ( ) 3!. j! k! ( j k)! The maximum is achieved whe j ad k are as close as possible to 3, ad this maximum is smaller tha c for some costat c. Hece P(S 2 = 0) c so P(S 2 = 0) < ad the radom walk is trasiet. d 4: The radom walk is still trasiet, sice the first 3 coordiates are trasiet. Exercise Fid a poit-recurret radom walk o R whose set of possible values is a coutable dese set i R. 3 2, 2 The Chug-Fuchs theorem Theorem 5 (Chug-Fuchs, 95) Let S be a radom walk i R d. The: (i) If d = ad S 0 i probability, the S is eighborhood-recurret. I particular, this happes if EX = 0. 4-3
(ii) If d = 2 ad S coverges i distributio to a cetered ormal distributio, the S is eighborhood-recurret. I particular, this happes if EX = 0 ad EX 2 <. (iii) If d = 3 ad the radom walk is ot cotaied i a plae the it is eighborhoodtrasiet (the coditio just meas that the set of eighborhood-possible values is ot cotaied i a plae). Remark 6 If the walk is o Z d the eighborhood-recurrece is the same as poitrecurrece ad the theorem still applies. Remark 7 If EX = µ the by the strog law of large umbers S µ almost surely. Therefore if EX exists ad is ozero, it s obvious that the radom walk is trasiet. We wo t prove the Chug-Fuchs theorem i its full geerality, but we will show some partial results, together with other criteria for recurrece. We will eed the followig theorem: Theorem 8 (Birkhoff ergodic theorem for fuctios of IIDs) Let X,X 2,... be IID radom variables i a state space S ad let g : S R be ay measureable fuctio. Defie Y = g(x,x +,...). If E Y < the almost surely ad i L. lim Y k = EY k= Remark 9 I fact, it is sufficiet to take the Y s to be a statioary ergodic sequece with E Y < We wo t prove this theorem (a proof ca be foud, for example, i []). However, we will use this result to prove the followig: Theorem 0 (Keste-Spitzer-Whitma rage theorem, 964) For a radom walk i R d let R = {x : k,s k = x} be the umber of distict poits visited, the almost surely. R P(o retur to 0) = lim 4-4
Remark This theorem also holds for ergodic markov chais ad eve statioary ergodic sequeces. Proof Note that Thus R R = [S j S k for k+ j ]. [S k ever revisited] = g(x k+,x k+2,...) for a appropriate measureable fuctio g. Now we ca use Birkhoff ergodic theorem ad get that R lim if Eg(X,X 2,...) = P(o retur to 0) almost surely. O the other had, fix M ad ote that R M Agai by Birkhoff we get that ( R lim sup limsup almost surely. But as M so it follows that ad we are doe. [S k ot visited agai by time k +M]+M. M + M = P(0 ot revisited by time M) [S k ot visited agai by time k+m] P(0 ot revisited by time M) P(0 is ever revisited), lim sup R P(0 is ever revisited) We are ow ready to prove part (i) of the Chug-Fuchs theorem, uder the additioal assumptios that the radom walk is o Z ad that EX exists (ad therefore EX = 0): Proof By the strog law of large umbers we kow that S 0 almost surely. Therefore for every ǫ > 0 we ca fid 0 (which is a radom variable by itself) such that S < ǫ for > 0. Hece for much larger tha 0 we get that R 3ǫ, so limsup R 3ǫ almost surely. Sice ǫ was arbitrary it follows that R 0, so by the Keste-Spitzer-Whitma theorem P(o retur to 0) = 0, ad the walk is recurret. ) = 4-5
3 Fourier aalytic criterio for recurrece For a radom walk o Z d, let ϕ(θ) = Ee iθ X be the characteristic fuctio of X. Note that ϕ is essetially from T d = [ π,π] d to C. Remider (i) Ee iθ S = ϕ (θ) (ii) We have the Fourier iversio formula: P(S = y) = (2π) d T d e iy θ ϕ (θ)dθ Theorem 2 A radom walk S o Z d is reucrret if ad oly if ( ) lim dθ = rր rϕ(θ) T d R Remark 3 (i) It is also true, but more difficult to prove, that S is recurret if ad oly if ( ) dθ = ϕ(θ) T d R (ii) Similarly, for geeral radom walks i R d, S is eighbourhood-recurret if ad oly if for some δ > 0 (ad the for every δ > 0) ( ) dθ = ϕ(θ) (this time ϕ is defied o R d ) ( δ,δ) d R (iii) If X has first ad secod momets, the (i dimesio): as θ 0. ϕ(θ) = +iθex θ2 2 EX2 +o(θ2 ) (iv) The weak law of large umbers holds if ad oly if ϕ (0) exists, ad the S µ i probability, where ϕ (0) = iµ (By a result of E. Pitma) Proof P(S = 0) = (2π) d T d ϕ (θ)dθ, 4-6
so for 0 < r < we have r P(S = 0) = (2π) d r ϕ (θ)dθ = (2π) d T d Here the fial step was Fubii s theorem, which applies because T d r ϕ (θ) dθ Sice the left had side is real it follows that ( r P(S = 0) = (2π) T d R d T d r dθ <. rϕ(θ) r ϕ (θ)dθ. T d ) dθ. Whe r ր the left had side coverges to P(S = 0), so we are doe. We would like to use this result for a few remarks about the relatio betwee recurrece ad momets Defiitio 4 The symmetric stable radom variable of idex α is the radom variable X o R such that Ee iθ X = e θ α Such a radom variable exists for every α 2. If α = 2 this is just the gaussia. If α < 2 this is a symmetric radom variable such that E X p < for every p < α, but E X α =. Note that as θ 0. Therefore ( δ,δ) d ϕ(θ) = e θ α θ α { dθ : α ϕ(θ) = < : α <, so these radom walks are recurret if ad oly if α. For α > this is a cosequece of the Chug-Fuchs theorem, sice EX = 0. For α = X has the Cauchy distributio, with desity π(+x 2 ). For such X, S is agai Cauchy distributed (as ca be see by calculatig its characteristic fuctio), so the weak law of large umbers does t hold eve though the walk is recurret. The recurrece coditio is ot oly about the tail of the radom varaible. I 964, Shepp gave examples of recurret walks with arbitrary heavy tails. I fact he proved the followig: 4-7
Theorem 5 For every fuctio ǫ(x) such that lim x ǫ(x) = 0, there exists a symmetric radom variable X such that P( X > x) ǫ(x) for all large x, ad the radom walk S is eighborhood (ad eve poit) recurret. Refereces [] Lalley S. Oe-dimesioal radom walks. http://galto.uchicago.edu/ lalley/courses/32/rw.pdf. 4-8