Simple Random Walk. Timo Leenman, June 3, Bachelor scription Supervisor: Prof. dr. F den Hollander

Transcription

1 Simple Radom Walk Timo Leema, Jue 3, 2008 Bachelor scriptio Supervisor: Prof. dr. F de Hollader

2 Cotets Defiitio of the radom walk 3 2 Recurrece of the radom walk 3 3 Rage of the radom walk 0 4 Probability measures ad stochastic covergece 5 5 Browia motio 8 Preface This treatise is o simple radom walk, ad o the way it gives rise to Browia motio. It was writte as my bachelor project, ad it was writte i such a way that it should serve as a good itroductio ito the subject for studets that have as much kowledge as I whe I bega workig o it. That is: a basic probability course, ad a little bit of measure theory. To that ed, the followig track is followed: I sectio, the simple radom walk is defied. I sectio 2, the first major limit property is studied: whether the walk be recurret or ot. Some calculus ad the discrete Fourier trasform are required to prove the result. I sectio 3, a secod limit property is studied: its rage, or, the umber of visited sites. I the full proof of the results, the otio of strog ad weak covergece presets itself, ad also the otio of tail evets. To uderstad these problems more precisely, ad as a ecessary preparatio for Browia motio, some measure theoretic foudatios are treated i sectio 4. Emphasis is put, ot o the formal derivatio of the results, but o the right otio of them i our cotext. I sectio 5, Browia motio is studied. First, i what maer simple radom walk gives rise to it, ad secodly its formal defiitio. Special care is devoted to explai the exact steps that are eeded for its costructio, for that is somethig which I foud rather difficult to uderstad from the texts I read o it. Timo Leema 2

3 Defiitio of the radom walk Radom walk describes the motio o a lattice, say Z d, of a walker that jumps at discrete time steps t, 2, 3,... to a ew, radomly chose, site. Such a radom walk ca be defied i various ways, resultig i various properties. With each time a radom variable is associated (the step), ad the distributio of these radom variables fixes the behaviour of the walk. Oe could for example defie a walk that ever visits a site twice, or oe that ever turs 80 degrees at oce. But the radom walk we are to look at here is simple radom walk. Its successive steps are chose idepedetly, they ca be of legth oly ad are chose uiformly out of the 2d possible directios o Z d. Formally, defie steps X i as radom variables o Z d as follows: Defiitio. The discrete radom variables X, X 2,... o Z d are called steps of the radom walk ad have the followig probability distributio: i N : P (X i e) 2d if e Zd ad e, ad P (X i e) 0 otherwise. Defiitio. S 0 0 Z d ad S X X for N is called the positio of the radom walk at time. So the radom walker begis his walk o the startig site S 0 0, ad by takig i.i.d. steps X i arrives at positio S at time. Because of the idepedece of the steps it is obvious that {S } N0 is a Markov process o the state space Z d, sice the future positios oly deped o the curret positio. Now that we have a Markov chai, we ca talk about trasitio probabilities. Our otatio shall be thus: call p(l) P (X l) P (S l) the oe-step trasitio probability, which is 2d for l a eighbour, ad 0 otherwise. Call P (l) P (S l) the -step trasitio probability, which equals the probability that the walk is at site l at time (startig i 0). 2 Recurrece of the radom walk I studyig the log term behaviour of the radom walk, oe of the first questios oe might be iterested i is whether the radom walker returs to its startig site. To this ed, defie F to be the probability that the radom walker evetually returs to the startig site S 0. If F, the the site S 0 is called recurret, if F <, the it is called trasiet. I the recurret case, it is obvious that the radom walker returs ot oly oce but ifiitely may times to S 0, whereas i the trasiet case, the radom walker may ever retur with positive probability F. I which latter case the umber of returs to S 0 is geometrically distributed with parameter F, ad therefore the expected umber of returs to a trasiet startig site is F F F. 3

4 The state space of a geeral Markov chai ca be partitioed ito recurret ad trasiet classes of states. I the simple walk, however, it is clear that all states commuicate (i.e. the walker has positive probability to reach ay give site startig from ay other give site), ad hece that it cosists of oly oe class. Therefore it makes sese to call the whole radom walk recurret or trasiet, wheever S 0 is so. The certaity of returig to every visited site o the oe had, ad the likelihood of ot returig to them o the other had, give recurret ad trasiet radom walks completely differet behaviour. A valuable result, due to Polýa, tells which simple radom walks are recurret: Polýa s Theorem. Simple radom walks of dimesio d, 2 are recurret, ad of d 3 are trasiet. The proof of this theorem is the object of this sectio. To that ed, we first eed a geeral criterio to see whether S 0 be recurret or ot. Theorem. The state S 0 0 is trasiet if ad oly if P (0) <. Proof. For t N, defie I t if S t 0, ad I t 0 otherwise. Note that N t I t is the umber of times that S 0 is revisited. For the expectatio of that umber the followig holds: [ ] E[N] E I t E[I t ] P (S t 0) P t (0). t t Computig the expectatio agai, we have t t E[N] kp (N k) [kp (N k) kp (N k + )] k P (N k) k k k F k, where the last equatio follows from the fact that every retur occurs idepedetly with probability F. Puttig the results together we get P t (0) E[N] t F k, k which diverges if F, ad coverges if F <. I other words, the radom walk is recurret precisely whe t P t(0) is ifiity. It is this equivalece that we will use to prove Polýa s theorem. First, we compute P (0). After that we compute the sum over to see 4

5 whether it diverges. There is a way to proceed that covers all dimesios at oce, amely, by givig a itegral expressio for P (0), ad aalyzig this expressio for. We will eed this procedure for dimesios d 3, but to illustrate that the situatio for d, 2 is sigificatly easier, we will first carry out a simple computatio for d, 2. d, 2 d : Imagie the oe-dimesioal lattice Z lyig horizotally. Ay path that the radom walker follows ca be uiquely represeted by a ifiite sequece (llrlrrrl...) (l stadig for left, r for right). Coversely, ay such sequece represets a path. Next, ote that ay path from 0 to itself has a eve legth, cotaiig as may steps to the left as to the right. Therefore P 2+ (0) 0; P 2 (0) ( 2 ) ( ) ( ) 2 2 (2)!!(2 )! 2 2. Now substitute Stirlig s approximatio of the factorial! e 2π as to get P 2 (0) 22 2 e 2 4π 2 e 2 2π So the ifiite sum becomes P (0) P 2 (0) π [ + o()] 2 2 as. π π [ + o()] > π, ad we see that the oe-dimesioal radom walk is recurret. d 2: For oe two-dimesioal walk, defie two oe-dimesioal walks i the followig way: 5

6 Let S be the two-dimesioal positio. Defie for every the positios of two oe-dimesioal walks S ad S 2 by the orthogoal projectio of S o the respective axes ad 2. The steps of a radom walk are the differeces betwee two successive positios, ad the two-dimesioal step X i ca take the values orth, east, south, west. The followig table gives the relatio betwee X i, X i ad X 2 i : N E S W X X 2 From this table it is obvious that the distributio of X give X 2 is the same as the margial distributio of X, ad P (X ) P (X ) P (X 2 ) P (X 2 ) 2. So i this way ay two oe-dimesioal radom walks correspod precisely to oe two-dimesioal radom walk, ad the other way roud. Therefore, i d 2 we ca write: P 2 (0) P (S 2 0) P (S2 0)P (S2 2 0) ( ) 2 π π ad, because still P 2+ (0) 0, the sum over becomes P (0) P 2 (0) π [ + o()]. So the two-dimesioal radom walk is recurret as well. d 3 The geeral method to prove recurrece or trasiece eeds some more computatio. The whole method is based o the well-kow theorem for Markov chais due to Chapma ad Kolmogorov, which i our otatio takes the form: Theorem. I the above otatio, the followig holds for all l Z d : P + (l) l Z d p(l l )P (l ). () I words this states that the probability of travellig to l i + steps ca be foud by summig over the positios the walker ca occupy at time. It is clear that the statemet uses the traslatio ivariace of the walk. The theorem ca be see as describig the evolutio of the walk: a recurrece relatio that expresses higher-step trasitio probabilities i terms of lower-step trasitio probabilities. The oe-step trasitio probabilities are prescribed by the defiitio of the radom walk. As may ordiary differetial equatios ca be solved by applyig the Fourier trasform F to them, 6

7 i a like maer we will solve our recurrece relatio usig the discrete Fourier trasform F, the properties of which are very much the same as of the cotiuous oe. It takes the form F(P (l)) P (k) l Z d e il k P (l), k [ π, π) d. For ease we defie the structure fuctio λ(k) as the Fourier traform of the oe-step trasitio probabilities: λ(k) p(k) l eil k p(l). We ca ow trasform equatio (), ad we get P + (k) l e il k l p(l l )P (l ). Note that e il k e i(l l ) k e il k is a costat (ot depedig o l ) that ca be placed behid the summatio sig. Therefore P + (k) p(l l )e i(l l ) k e il k P (l ). l l Now call m l l, ad the above equals p(m)e im k e il k P (l ) λ(k) P (k) P + (k). l m+l The recurrece relatio has ow become easy to solve: we oly eed the iitial coditio. This is P 0 (l) δ 0l (at t 0 the walker must be i the origi), ad i Fourier trasform this coditio is P 0 (k) l eil k δ 0l (oly for l 0 does the delta-fuctio ot vaish). Substitutig this iitial coditio, we see that the solutio of () becomes P (k) λ(k). The iverse Fourier trasform has the form of the d-dimesioal itegral P (l) F ( P (k)) (2π) d... e il k P (k)dk. k [ π,π) d So the formal solutio (for ay dimesio) of the trasitio probabilities P is P (l) (2π) d... e il k λ(k) dk. k [ π,π) d Util ow the calculatio was purely formal. If we wat to test the walk for trasiece, we must sum the trasitio probabilities P (0)over. For that, we eed to kow λ(k) ad the evaluate the multiple itegral. Because we are oly iterested i whether the sum P (0) coverge or ot, it suffices for our purpose to approximate λ(k), ad the to determie the limitig behaviour of P (0) for : if it approaches 0 fast eough, the the sum will coverge, ad hece the radom walk will be trasiet. 7

8 So, what is λ(k) for our simple radom walk? The aswer follows from a short computatio. Write the vector k (k, k 2,..., k d ) T, ad fill i the fomula for the structure fuctio: λ(k) p(k) l e il k p(l) l: l e il k 2d. Here we use that the oe-step trasitio probability p(l) equals 2d if l is a uit vector, ad equals 0 otherwise. Deote these 2d uit vectors by e j ad e j, for j,..., d. The λ(k) 2d 2d d d j d j [ e ie j k + e i( e j k) ] 2d d j [cos(k j ) + cos( k j )] + i 2d d cos(k j ). j [ ] e ik j + e ik j d [si(k j ) + si( k j )] For the sake of approximatig this with a fuctio that is more easily itegrated over, recall the followig two Taylor expasios for x 0: j cos(x) x2 2! + h.o. ad ex + x x2 2! + h.o., where h.o. meas higher order terms. Substitute this ito the formula for λ(k) to obtai λ(k) d d j ( k2 j 2! + h.o.) 2d k 2 + h.o. e 2d k 2 for k 0. The itegral we are to calculate has ow become P (0) (2π) d... e il k e 2d k 2 dk for k 0 k [ π,π) d but this approximatio oly holds for small k. Yet this is all we eed, because we are iterested i the limitig behaviour as, i which case the secod factor i the itegrad clearly becomes very small, uless k is take to be so small as to compesate for the large. Thus it ca be see that the domiatig ( cotributio to the itegral is cotaied i the regio where k O ). I the limit this meas that k approaches 8

9 0, ad the value of the itegral is ot affected whe we itegrate over the whole of R d i stead of oly [ π, π) d. Hece P (0) (2π) d... e 2d k 2 dk for. R d Now observe that the itegrad oly depeds o the legth of k. This suggests a trasformatio to spherical coordiates, because the the itegrad oly depeds o the radius r k. The above itegral equals (2π) d 0 d dr Bd r e 2d r2 dr, where Br d is the volume of the ball of radius r i d dimesios. For computig Br d, defie the stepfuctio θ [0, ). The, for all d, Br d θ(r 2 x 2 )dx. R d Now substitute y x r, ad hece x 2 r 2 y 2, ad dx r d dy. This yields Br d θ(r 2 r 2 y 2 )r d dy r d θ( y 2 )dy ω d r d, R d R d where ω d represets the volume of the uit sphere i d dimesios. Because d dr Bd r dr d ω d, our asymptotic itegral expressio for P (0) becomes (2π) d dω d r d e 2d r2 dr. 0 To evaluate further, substitute x 2d r2. The dx dr d r, so rdr d dx, ad ( ) d 2 ( ) 2d 2d r d 2 x 2 d x 2 d, ad the boudaries 0 ad remai the same. Now the itegral has become, for, ) 2 d P (0) x 2 d e x d dx (2π) d 0 dω d ( 2d (2π) d ω d(2d) 2 d Γ ( ) 2 d C(d) 2 d, 2 d where Γ(a) 0 e x x a dx for a > 0, ad C(d) is a costat that depeds o the dimesio. At last we ca sum over the trasitio probabilities. Note that C(d) C(d) < if d 3. d 2 d 2 Because for large eough P (0) 2 C(d), we coclude that for d 3 2 d it must hold that P (0) <, ad therefore simple radom walk i dimesio d 3 is trasiet. 9

10 3 Rage of the radom walk Ituitively, it is clear that a trasiet radom walker is much more likely to visit ew sites of the lattice tha a recurret oe. To make this precise, defie the rage R of a radom walk at time as the umber of distict poits visited withi time : Defiitio. 0 : R card({0 S 0, S,..., S }). Defiig F to be the probability that the walker will evetually retur to its startig site S 0, the behaviour of the rage is stated i the followig theorem: Theorem. ɛ > 0 : lim P ( R ( F ) > ɛ) 0. Proof. Defie φ 0 ad, for k N φ k if S i S k for all i,.., k ad φ k 0 otherwise. I other words, φ k if ad oly if the radom walker visits a ew site o its k th step. It is obvious that R k0 φ k, ad because of liearity of expectatios also E[R ] k0 E[φ k] holds. For ay k N we ca write: E[φ k ] P (φ k ) P (S k S k, S k S k 2,..., S k S 0 0) P (S k S k 0, S k S k 2 0,..., S k 0) P (X k 0, X k + X k 0,..., X k X 0) P (X 0, X + X 2 0,..., X X k 0) P (S j 0 for j,..., k) k F j (0, 0), j where i the fourth lie we reverse the idices, ad i the sixth lie F j (0, 0) is the probability that the radom walker, startig i 0, returs to 0 for the first time o its j th step. Takig the limit k, we get lim E[φ k] F k ad this equals 0 if ad oly if the radom walk is recurret. Cosequetly, lim E[φ k ] lim E[R ] F. k0 To proceed, we cosider the recurret ad the trasiet case separately. For ɛ 0, write ( ) R P > ɛ P (R k) k ɛ P (R k) k:k>ɛ ɛ k:k>ɛ kp (R k) ɛ E[R ]. k0 0

11 For the recurret case it therefore follows that lim P ( R > ɛ) 0 for all ɛ > 0, ad we are doe. The trasiet case is more complicated. First, otice that for ay cotious radom variable X o R 0 the followig holds: E[X] 0 xp (X x)dx ɛ 0 xp (X x)dx + ɛ xp (X x)dx ɛ ɛ f(x)dx ɛp (X > ɛ) where f(x) is the prbability desity fuctio, ad this iequality also holds for discrete radom variables o R 0. Secodly, use this iequality for the radom variable R ( F ) 2. Startig from the probability we are iterested i, we get ( ) R P ( F ) > ɛ P ( R ( F ) 2 > 2 ɛ 2) 2 ɛ 2 E[ R ( F ) 2 ] 2 ɛ 2 E[R2 2( F )R + 2 ( F ) 2 ] 2 ɛ 2 E[R2 2R E[R ] + 2(E[R ]) 2 2( F )E[R ] + 2 ( F ) 2 ] 2 ɛ 2 E[R2 2R E[R ] + (E[R ]) 2 ] + 2 ɛ 2 (2 ( F ) 2 2( F )E[R ] + (E[R ]) 2 ) 2 ɛ 2 E[(R E[R ]) 2 ] + ( [ ]) 2 R ɛ 2 F E. Write E[(R E[R ]) 2 ] var(r ). To prove that the above probability coverges to zero as, it suffices to prove that lim var(r 2 ) 0, because it was already show that lim E[ R ] F, so that the secod term of the expressio vaishes i the limit. Cotiue by computig var(r ) as follows (usig the liearity of expectatios): var(r ) E[(R ) 2 (E[R ]) 2 ] E E 2 j0 k0 j0 k0 φ j φ k E j0 j0 φ j (E[φ j φ k ] E[φ j ]E[φ k ]) 0 j k φ j (E[φ j φ k ] E[φ j ]E[φ k ]) + k0 φ k [ ] E φ k k0 E j0 φ j E[φ j E[φ j ]]. j0 2

12 This last equality follows from the fact that summig over the elemets of a symmetric (square) matrix, oe may as well take twice the sum over the elemets uder the diagoal, ad add the diagoal elemets (otice that φ 2 j φ j). Because E[φ j E[φ j ]] E[φ j ], var(r ) ca be estimated by var(r ) 2 0 j k (E[φ j φ k ] E[φ j ]E[φ k ]) + E[φ j ]. But we ca estimate it by a yet simpler expressio: Notice that for 0 j < k, E[φ j φ k ] P (φ j φ k ) P (S j S α for 0 α < j, S k S β for 0 β < k) j0 P (S j S α for 0 α < j, S k S β for j < β < k) P (X j 0, X j + X j 0,..., X j X 0; X k 0, X k + X k 0,..., X k X j+ 0) P (X 0,..., X X j 0) P (X 0,..., X X k j 0). The factorizatio ad mixig of idices is allowed because the X i are i.i.d. Now recall that E[φ k ] P (X 0,..., X X k 0), so the iequality says that E[φ j φ k ] E[φ j ]E[φ k j ] for 0 j < k. Substitutio ito the former estimate of var(r ) yields var(r ) 2 E[φ j ] (E[φ k j E[φ k ]) + E[R ] j0 kj+ Sice E k [φ k ] k j F j(0, 0), we have {E[φ k ]} k0 is a mootoe oicreasig sequece. But for ay such sequece a a 2... a the sum (a k j a k ) (a + a a j ) (a j+ + a j a ) kj+ is maximized by takig j 2 (that is, roud off 2 dowward). Ideed, by takig j smaller, the left term icreases less tha the right oe ad, by takig j larger, the left term decreases more tha the right oe. Its maximum value is (a a 2 ) [(a a ) a a 2 ]. Takig a k E[φ k ], ad recallig that k0 E[φ k] E[R ], it therefore holds that var(r ) 2 j0 E[φ j ]E[R 2 + R 2 + R ] + E[R ]. Because we already showed that lim E[ j0 φ j] lim E[ R ] F, we get [ ] E[R lim 2 var(r ) 2( F ) lim E 2 + R 2 + R + 0 2

13 ( F 2( F ) 2 + F 2 ) ( F ) 0, which was still left to be proved. The above theorem states that the radom variable R coverges to F R i probability, but i fact a stroger statemet also holds: coverges to F almost surely. The proof thereof requires the ergodic theorem, which we caot prove here. The differece betwee these two types of covergece of radom variables is defied i sectio 4, but ituitively it meas the followig. Cosider the collectio of all possible paths (of ifiite legth) a radom walker might take. For each of those paths with every time a value R is associated. Strog covergece meas that, if oe of those paths is selected, the it is with probability oe a path for which the sequece ( R ) N coverges to F. If some arbitrary deviatio ɛ 0 is give, the the probability (whe selectig a path) p P ( R ( F ) > ɛ) depeds o. Covergece i probability meas that that the sequece (p ) N coverges to 0. Theorem. P (lim R F ). Proof. We will defie two sequeces (D ) N ad (R,M ) N such that D R R,M, ad use these to determie the value of the limit. First, defie R,M like R, but at every M th step forget which sites were already visited (but do ot reset the couter to 0). Formally, defie the rages of subsequet M-step walks ad ow add these up to get Z k (M) card{s km, S km+,..., S k(m+) }, R,M M k0 Z k (M), which is the sequece described above. It is obvious that R,M R. Note R that it is ot clear yet that lim exists at all, so that we will estimate dowwards by takig the limit iferior ad upwards by takig the limit superior (because these exist for every sequece). Thus it must hold that lim sup R lim sup M k0 Z k(m). 3

14 Now, the Z k (M) are i.i.d., ad so the strog law of large umbers ca be applied to obtai lim sup M M M k0 Z k (M) M E[Z 0(M)] M E[R M] a.s. Now take the limit M. We already saw that M E[R M] coverges to F, ad therefore we get lim sup R F a.s. Secodly, defie D as the umber of distict sites visited i time that the walker ever visits agai. Formally, let D k0 ψ k with ψ k if X k X k+i 0 for all i N ad ψ k 0 otherwise. So ψ k oly if after the k th step the walker ever returs to the site where it is at time k, i.e. if the walker visits the curret site for the last time. The umber of times that the walker visits a site for the last time is evidetly smaller tha the umber of sites visited, so D R. Y 0, Y,... is said to be a statioary sequece of radom variables if, for every k, the sequece Y k, Y k+,... has the same distributio, that is, for every, the ( + )-tuples (Y 0, Y,..., Y ) ad (Y k, Y k+,..., Y k+ ) have the same joit probability distributio. I particular, the Y i s are idetically distributed, but they may be depedet. Because of symmetry ad the Markov property of the simple radom walk, it is clear that (ψ k ) k N is a statioary sequece. Therefore, by the ergodic theorem the limit lim k0 D ψ k lim exists with probability (but it may still be a radom variable). But D lim assumig a certai value is a so-called tail evet. Ituitively, tail evets are those evets for a sequece of radom variables that would still have occurred if some fiite umber of those radom variables would have had a differet realisatio. Tail evets are treated more thoroughly i D sectio 4. Ideed, all evets of the form lim D < C, lim C etc., are tail evets, ad must therefore, accordig to Kolmogorov s 0- law, 4

15 occur with probability 0 or. Cosequetly, the limit must eeds be equal to a costat. This costat ca oly be the atural cadidate, amely, lim ψ k E[ψ 0 ] P (X X i 0 for all i ) F a.s. k0 Cosequetly, lim if R lim if D lim D F a.s. R Fially, ote that the first statemet (lim sup F a.s.) ad R the secod statemet (lim if F a.s.) together imply the R statemet of the theorem (lim F a.s.). 4 Probability measures ad stochastic covergece The purpose of this sectio is to defie more precisely what is meat by radom variables ad their covergece. This is doe i measure-theoretic terms, because that is the oly way to make precise our costructio of socalled Browia motio i sectio 5. While this treatise is o simple radom walk, ad ot o measure-theoretic probability, we will put more emphasis o the ituitive iterpretatio of the defiitios tha o the proof of their properties, which ca be foud i []. We use a probability space to model experimets ivolvig radomess. A probability space (Ω, Σ, P) is defied as follows: Sample space. Ω is a set, called the sample space, whereof the poits ω Ω are called sample poits. Evets. Σ is a σ-algebra of subsets of Ω, that is, a collectio of subsets with the property that, firstly, Ω Σ; secodly, wheever F Σ the F C Ω \ F Σ; thirdly, wheever F Σ( N), the F Σ. Notice that these imply that Σ cotais the empty set, ad is closed uder coutable itersectios. All F Σ are called measurable subsets, or (whe talkig about probability spaces) evets. Probability. P is called a probability measure o (Ω, Σ), that is, a fuctio P : Σ [0, ] that assigs to every evet a umber betwee 0 ad. P must satisfy: firstly, P( ) 0 ad P(Ω) ; secodly, wheever (F ) N is a sequece of disjoit evets with uio F F, the P(F ) N F (σ-additivity). 5

16 Radomess is cotaied i this model i the followig way: Whe performig a experimet, some ω Ω is chose i such a way that for every F Σ, P(F ) represets the probability that the chose sample poit ω belogs to F (i which case the evet F is said to occur). Some statemet S about the outcomes is said to be true almost surely (a.s.), or with probability, if F {ω : S(ω) is true} Σ ad P(F ). If R is a collectio of subsets of S, the σ(r), the σ-algebra geerated by R, is defied to be the smallest σ-algebra cotaied i Σ. The Borel σ-algebra B is the smallest σ-algebra that cotais all ope sets i R. A fuctio h : Ω R is called Σ-measurable if h : B Σ, that is, if h(a) Σ for all A B. (Compare cotiuous maps i topology: a map is called cotiuous if the iverse image F (G) is ope for all ope G. A map is called measurable if the iverse image of every measurable subset is measurable.) Give (Ω, Σ), a radom variable is a Σ-measurable fuctio. So, for a radom variable X: X : Ω R ad X : B Σ Give a collectio (Y γ ) γ C of maps Y γ : Ω R, its geerated σ-algebra Y σ(y γ : γ C) is defied to be the smallest σ-algebra Y o Ω such that each map Y γ is Y- measurable (that is, such that each map is a radom variable). The followig holds: Y σ(y γ : γ C) σ({ω Ω : Y γ (ω) B} : γ C, B B). If X is a radom variable for some (Ω, Σ), the obviously σ(x) Σ. Suppose (Ω, Σ, P) is a model for some experimet, ad that the experimet has bee performed, that is, some ω Ω has bee selected. Suppose further that (Y γ ) γ C is a collectio of radom variables associated with the experimet. Now cosider the values Y γ (ω), that is, the observed values (realisatios) of the radom variables. The the ituitive sigificace of the σ-algebra σ(y γ : γ C) is that it cosists precisely of those evets F for which it is possible to decide whether or ot F has occurred (i.e. whether or ot ω F ) o the basis of the values Y γ (ω) oly. Moreover, this must be possible for every ω Ω. Give a sequece of radom variables (X ) N ad the geerated σ- algebras T σ(x +, X +2,...), defie the tail σ-algebra T of the sequece (X ) N as follows: T N T. 6

17 So T cosists of those evets which ca be said to occur (or ot to occur) o the basis of the realisatios of the radom variables, beyod ay fiite idex. For such evets the followig theorem states, that they will either occur with certaity, or ot at all. Kolmogorov s 0- law. Let (X ) N be a sequece of idepedet radom variables, ad T the tail σ-algebra thereof. The F T P(F ) 0 or P(F ). Suppose X is a radom variable carried by the probability space (Ω, Σ, P). We have Ω X R [0, ] P Σ X B [0, ] P σ(x) X B. Defie the law L X of X by L X P X, so L X : B [0, ]. The law ca be show to be a probability measure o (R, B). The distributio fuctio of a radom variable X is a fuctio F X : R [0, ] defied by: F X (c) L x (, c) P(X c) P({ω : X(ω) c}). Because we have defied a radom variable as a fuctio from Ω to R, we ow have several otios of a covergig sequece of radom variables. The usual modes of covergece for fuctios are all well-defied for ay sequece (X ) N of radom variables, ad we may for example cosider uiform covergece or poitwise covergece to some radom variable X. The latter oe is weaker, ad we mea by it: ω Ω : lim (X (ω)) X(ω) or, shortly, X (ω) X(ω). But i practice, for radom variables we are oly iterested i yet weaker modes of covergece, which we defie below. A sequece (X ) N of radom variables is said to coverge to X: almost surely (or, with probability ), if P({ω Ω : X (ω) X(ω)}). Note that ot for all ω Ω eeds (X (ω)) N coverge to X(ω), but the set of ω s for which it does coverge, has probability oe. Which is the same as sayig that if for some radom ω Ω, the sequece of real umbers (X (ω)) N is cosidered, it is certai to coverge to X(ω). i probability, if ɛ > 0 : P({ω Ω : X (ω) X(ω) > ɛ}) 0. Note that (X (ω)) N may ot coverge to X(ω) for all ω Ω, but for ay ɛ > 0 the probability that X deviates from X more tha ɛ for a fixed teds to 0 as. 7

18 i distributio, if F X (c) F X (c) for all cotiuity poits c of F X. Which expresses othig but the poitwise covergece of the distributio fuctios, ad tells othig about the radom variables themselves. Covergece almost surely is also called strog covergece ad is deoted as, covergece i probability is also called weak covergece ad is deoted P, covergece i distributio is also called covergece i law ad is deoted. Note that for covergece i law, the sequece of radom variables eeds ot be defied o the same probability space as its limit. I particular, the X s may be defied o a discrete space, while X may be defied o a cotiuous space. For example, if P(X i ) for i, 2,..., the X X, with X uiformly distributed o [0, ]. It ca be show that strog covergece implies weak covergece, ad weak covergece implies covergece i law, but oe of the three are equivalet i geeral. 5 Browia motio Imagie the followig: istead of executig a radom walk o the stadard lattice with site distace at time steps of size, we make the lattice ever arrower ad reduce our time steps appropriately. Evetually we will get some radom process i cotiuous space ad time. Loosely speakig, makig the lattice arrower ad reducig the time steps should happe i some harmoized maer: whe the distace travelled per step teds to 0, the umber of steps per time uit should ted to ifiity i the right way to have the proportio of the visited area be costat. We proceed to show how the above idea ca be made precise. ( For t 0, cosider the sequece S t. This is a sequece ) N of positios, rescaled such that it coverges to a radom variable as, by the cetral limit theorem. To apply this theorem, we must kow expectatio ad variace of the steps X i. Obviously E[X i ] 0, ad so var(x i ) E[Xi 2] E[X i] 2 E[Xi 2] 2d 2d i 2. With E[X i ] µ ad var(x i ) σ 2 the cetral limit theorem states ad therefore, for µ 0 ad σ, i i X i µ σ N(0, ), X i i X i S N(0, ). 8

19 Now fix t 0 ad observe that S t S t t t S t N(0, ). t Multiplyig by t we ca ow see that S t must for t 0 coverge i distributio to the ormal distributio N(0, t) of expectatio 0 ad variace t. Because of the idepedece of the steps X i, it is ( also clear that for) ay t 0 ad s 0 with t s 0, it must hold that S t S s N(0, t s). Recall that covergece i distributio does ot mea covergece of the sequece of actual values whe a experimet is performed (i fact, the limit lim S t does ot exist with probability ). Therefore it is impossible to treat the desired cotiuous process as a actual limit of some rescaled radom walk, but the covergece i distributio strogly suggests a defiitio of the limitig process by usig the acquired ormal distributios, that is, a process with idepedet icremets that are ormally distributed. But first, let us defie precisely what we mea by a cotiuous process, ad state some of its properties. A stochastic process X is a parametrized collectio of radom variables (X t ) t T defied o a probability space (Ω, Σ, P ) ad assumig values i R d. For our process, T [0, ), ad hece it is called cotiuous. The fiite-dimesioal distributios of a cotiuous process X are the measures µ t,...,t k defied o (R d ) k by µ t,...,t k (B,..., B k ) P (X t B,..., X tk B k ), where B i are Borel sets, ad t i T, for i,..., k. I other words, the fiitedimesioal distributios are the joit laws of the fiite collectios of radom variables out of the cotiuous process. Properties like cotiuity of the paths of the process are therefore ot determied by the fiite-dimesioal distributios, ad hece it is clear that a process X is ot equivalet to its fiite dimesioal distributios. Coversely, give a set {µ t,...,t k : k N, t i T fori,..., k} of probability measures o (R d ) k, uder what coditios ca a stochastic process be costructed, that has µ t,...,t k as its fiite-dimesioal distributios? Sufficiet coditios are give i the followig theorem. Kolmogorov s extesio theorem. For t,..., t k T, let µ t,...,t k be probability measures o (R d ) k such that 9

20 ) µ tσ(),...,t σ(k) (B... B k ) µ t,...,t k (B σ ()... B σ (k)) for all permutatios σ o {, 2,..., k} ad all t i T, i,..., k; 2) µ t,...,t k (B,..., B k ) µ t,...,t k,t k+,...,t k+m (B... B k (R d ) m ) for all t i T, i,..., k; The there exists a probability space (Ω, Σ, P ), ad a cotiuous process (X t ) t T o Ω, such that for all E i R d, i,..., k: µ t,...,t k (E... E k ) P (X t E,..., X tk E k ). This gives us the existece of some process, whereof oly the fiite dimesioal distributios are kow. It tells us othig about the shape of Ω (which, of course, is ot uique), yet the theorem is of crucial importace, sice it allows us to cosider joit laws of ifiite collectios of radom variables draw out of the process, ad therefore questios o cotiuity of paths, etc. Our cosideratio of the rescalig of the radom walk yielded very atural ( cadidates for a collectio of measures o (R d ) k, amely, those to which S t S s coverge i probability. The procedure for the ) N formal defiitio of the desired process comprises therefore, firstly, the defiitio of such a collectio of probability measures o (R d ) k ad secodly, the applicatio of the extesio theorem. This is carried out below. Defie p(t, y) : R 0 R d R as the joit probability desity fuctio of d idepedet ormal radom variables with variace t 0: p(t, y) (2πt) d 2 e y 2 2t for y R d, t 0. I order to defie the required probability measures o (R d ) k, first defie for k N ad 0 t... t k the measure P t,...,t k by P t,...,t k (E... E k ) p(t, x )p(t 2 t, x 2 x )... p(t k t k, x k x k )dx...dx k, E... E k where as a covetio p(0, y)dy δ 0 to avoid icosistecies. Secodly, exted this defiitio to all fiite sequeces t,..., t k by usig the first coditio i Kolmogorov s extesio theorem. The also the secod coditio is satisfied, because p is a probability desity (that itegrates to ). So there exists a probability space (Ω, Σ, P ) ad a cotiuous process (B t ) t 0 o Ω such that the fiite-dimesioal distributios of B t are give by the prescribed oes. This process (B t ) t 0 is called Browia motio. The fact that it has idepedet ad ormally distributed icremets ca be easily see from our defiitio. The third essetial property of Browia motio is its cotiuity. To show this, we use aother theorem of Kolmogorov. 20

21 Kolmogorov s cotiuity theorem. Let X (X t ) t 0 be a cotiuoustime process. If for all T > 0 there exist α, β, D > 0 such that E [ X t X s α ] D t s +β for 0 s, t T, the the paths of X are cotiuous with probability, that is, P (t X t is cotiuous). We will use this result to show cotiuity of Browia motio i dimesio d. Because (B t B s ) N(0, t s), partial itegratio gives E [ B t B s α ] Take α 4 to get E[ B t B s 4 ] x α e t s 2π x2 2(t s) dx α t s 2π x α 2 e 2 2(t s) t s 2π α 2 2(t s) α 3 2 2(t s) t s 2π 3(t s)(t s) x2 2(t s) dx x α 4 e π ( 2(t s) 3(t s)2 t s 2π 2(t s)π 3(t s) 2. x2 2(t s) dx. Hece for α 4, D 3, β, Browia motio satisfies the cotiuity coditio E [ B t B s α ] D t s +β. Although the Browia motio as we defied it above is ot uique, we do kow that for ay ω Ω the fuctio t B t (ω) is cotiuous almost surely. Thus, every ω Ω gives rise to a cotiuous fuctio from [0, ) to R d. I this way, we ca thik of Browia motio as a radomly chose elemet i the set of cotiuous fuctios from [0, ) to R d accordig to the probability measure P. Or, i our cotext: a radom cotiuous path startig at the origi. ) Refereces [] Probability with Martigales, D. Williams, Cambridge Uiversity Press, 99. [2] Stochastic Differetial Equatios, B. Øksedal, Spriger-Verlag, 985. [3] Pricipals of Radom Walk, F. Spitzer, Spriger-Verlag, secod editio,

22 [4] Radom Walks ad Radom Eviromets, B. Hughes, Oxford Uiversity Press, 995. [5] Probability: Theory ad Examples, R. Durret, Duxbury Press, secod editio,