Lecture 12: Subadditive Ergodic Theorem

Transcription

1 Statistics 205b: Probability Theory Sprig 2003 Lecture 2: Subadditive Ergodic Theorem Lecturer: Jim Pitma Scribe: Soghwai Oh 2. The Subadditive Ergodic Theorem This theorem is due to Kigma ad also kow as Kigma s subadditive ergodic theorem [4]. Let (Ω, F, P, T ) be a probability space with a measurable map T : Ω Ω which preserves P. The set Ω is defied such that there is a two parameter family of radom variables, (X(m, ), 0 m < < ). Each X(m, ) is itegrable with respect to P for 0 m < <. If X(m, ) satisfies: I. X(m +, + ) = X(m, ) T ad II. X(0, ) X(0, m) + X(m, ), the there exists a a.s. limit where Y [, ) ad Y is T -ivariat. X(0, ) lim = Y (2.) Notice that Y ca be. With additioal assumptios as show i Durrett [], we ca say more about Y. For istace, if T is ergodic, the Y is a costat. 2.2 Proof The followig proof is due to Steele [5]. The first step is to cosider X(m, ) = X(m, ) k=m+ X(k, k). (2.2) The first term is subaddtive while the secod term is additive so X satisfies the subadditivity coditios (I) ad (II). Sice the subadditivity coditio (II) holds, X(0, 2) = X(0, 2) X(0, ) X(, 2) 0 X(0, 3) = X(0, 3) X(0, ) X(, 2) X(2, 3) X(0, 3) X(0, 2) X(2, 3) 0... (2.3) 2-

2 2-2 Lecture 2: Subadditive Ergodic Theorem Thus X(, ) 0. Hece X(0, ) = X(0, ) X(k, k) k= X(0, ) = X(0, ) X(0, ) = X(0, ) + X(k, k) k= X(k, k) k= X(0, ) = X(0, ) + X(0, ) T k. (2.4) k=0 By Birkhoff s ergodic theorem the last sum i (2.4) coverges a.s. to a limit E(X(0, ) I), where I is the T -ivariat σ-field. So it is eough to show that X(0, )/ has a a.s. limit ad, without loss of geerality, we ca assume X(, ) 0. Lemma 2. Let Y = lim if X(0, ) where X(m, ) is defied as i the theorem. The Y is a.s. ivariat, i.e. Y = Y T a.s. Proof: Observe that X(0, + ) + = X(0, + ) + + { X(0, ) X(0, ) T + } X(0, ) T (2.5) sice X(0,) + ad 0. Thus lim if X(0, ) X(0, ) T lim if Y Y T for all ω Ω. (2.6) The for ay ratioal α, we have (Y > α) (Y T > α) = T (Y > α). But P (Y > α) = P (Y T > α) so (Y > α) = (Y T > α) a.s. for all ratioal α. Hece Y = Y T a.s. Now fix ɛ > 0 ad M > 0 ad let Y M = Y ( M) where Y is defied i Lemma 2. ad (x y) = max(x, y). Defie two sets as followig: B M (L) = { ω Ω : X(0, l)(ω) > l(y M (ω) + ɛ) for all l L } G M (L) = B M (L) C = { ω Ω : l L s.t. X(0, l)(ω) l(y M (ω) + ɛ) }. We the do a recursive cuttig up of the orbit for each ω Ω. We decompose [0, ) ito three types of sets as show i Figure 2.. The set E is a collectio of sigletos σ j where T σj (ω) B M (L) for j v. The set F is a collectio of itervals [a i, a i + l i ) for i u such that T ai (ω) G M (L), i.e. X(0, l i ) T ai (ω) l i (Y M (T ai (ω)) + ɛ). Lastly, the set H cotais the remaiig sigletos i [ L, ). Let w = H. Notice that u, v, w are r.v. s.

3 Lecture 2: Subadditive Ergodic Theorem 2-3 Figure 2.: A orbit of ω, {ω, T (ω), T 2 (ω),...} Usig the subadditivity, X(0, ) u X(a i, a i + l i ) + u X(a i, a i + l i ) ( u ) l i )(Y M + ɛ v X(σ j, σ j+ ) + j= s= w X(s, s + ) ( u ) Y M l i + ɛ, (2.7) where the secod iequality holds sice the last two summatios are less tha 0 (recall that X(, ) 0). O the other had for all ω, u(ω) So by Birkhoff s ergodic theorem, l i (ω) lim if BM (L) T i (ω) L. (2.8) u l i E( BM (L) I) = P(B M (L) I) = P(G M (L) I). (2.9) Now because of the way we have defied B M ad G M, for ay fixed M > 0, P(G M (L) I) as L. Combiig it with (2.7) gives lim sup X(0, ) Y M P(G M (L) I) + ɛ a.s. Y M + ɛ as L Y as M, ɛ 0 (2.0) ad the proof is complete.

4 2-4 Lecture 2: Subadditive Ergodic Theorem 2.3 Examples 2.3. Birkhoff s Ergodic Theorem If X(m, ) = j=m X T j where X is a itegrable r.v., the X(0, ) = X(0, m) + X(m, ) satisfyig the coditios for the subadditive ergodic theorem. The the coclusio from the subadditive ergodic theorem is exactly the same as that of Birkhoff s ergodic theorem Percolatio Cosider a iteger lattice Z d i which edges x, y Z d are coected if x y = (see Figure 2.2). Each edge e is associated with a r.v. T e. T e is the time take for a message to travel across the edge e. Let X(0, ) be the shortest time take for a message to travel from 0 to v, where v = (, 0, 0,..., 0). The X(0, ) = mi path 0 v e path T e (2.) ad X(m, ) = mi path m 0 v e path T e. (2.2) Sice there may be a miimum path from 0 to v without visitig m v, X(0, ) X(0, m) + X(m, ). (2.3) Now assume T e are idepedet r.v. s with a idetical distributio. So Ω = {T e, e E}, where E is a set of edges i Z d. Defie a map T : Ω Ω such that if ω = (t e, e E) the T (ω) = (t e +, e E), where e + is the edge o the right of e with the same orietatio (see Figure 2.2). The subadditive ergodic theorem X(0,) tells that if T is ergodic the there is a costat limit Y = lim, meaig that a message moves at a costat speed. For more about percolatio, see Grimmett [2]. Figure 2.2: Z 2 lattice

5 Lecture 2: Subadditive Ergodic Theorem Logest Icreasig Subsequece The problem of studyig the behavior of the logest icreasig subsequece of a radom permutatio is kow as a problem of Ulam ad Hammersley[3] was first to study this problem aalytically usig the subadditive ergodic theorem. Let σ be a permutatio of {, 2,..., } ad assume that all! permutatios are equally likely. A icreasig subsequece of a permutatio σ is a subsequece (σ(k ), σ(k 2 ),..., σ(k l )) such that k < k 2 < < k l ad σ (k ) < σ (k 2 ) < < σ (k l ), where l is the legth of the icreasig subsequece. Let L be the legth of the logest icreasig subsequece. For example, if we are give a set {, 2, 3, 4, 5} ad its permutatio σ () =, σ (2) = 3, σ (3) = 5, σ (4) = 2, σ (5) = 4, the icreasig subsequeces are (, 2, 4), (, 3, 5), (, 3, 4), (, 2),.... Thus L 5 = 3. The mai difficulties with this problem is that it is hard to give ay reasoable expressio for either EL or V ar(l ) ad it is otrivial to fid the asymptotic behavior of L as teds ifiity. Hammersley [3] has show the existece of a a.s. limit of L / by igeiously trasformig the problem ito a uit rate Poisso process o a two dimesioal plae ad usig the subadditive ergodic theorem. Later it has bee show that lim L / = 2 a.s. See Durrett [] for the refereces o this topic. Istead of workig with radom permutatios, we may as well work with a sequece of i.i.d. r.v. s with a cotiuous distributio. Let X, X 2,... be a sequece of i.i.d. r.v. s with uiform distributio o [0, ]. Let ˆL be the legth of the logest icreasig subsequece i X, X 2,..., X. The ˆL = d L sice there is o tie a.s. (each X has a cotiuous distributio) ad the raks of X, X 2,..., X correspods to a permulatio of {, 2,..., }. By usig i.i.d. r.v. s we have radomized the vertical scale as show i Figure 2.3. Figure 2.3: A istace of X, X 2,..., X 7 Uif[0, ] i.i.d. ad its logest icreasig subsequece (i circle), L 7 = 4 Now cosider a Poisso poit process i (0, ) (0, ) with uit itesity per uit area. See Figure 2.4. Take a k k box. Let Mk be the legth of the logest iceasig subsequece of poits i the k k box. Let N k be the umber of poits i the k k box. Say N k =. The vertical values are i.i.d. uiform o [0, k] like the previous example with uiform i.i.d r.v. s. Hece (Mk N k = ) = L i distributio. By superadditivity, Mk+m M k + M m T k, where T k shifts the origi from (0, 0) to (k, k) ad preserves the measure (see Figure 2.4). Now apply the subadditive ergodic theorem to M ad coclude that where Y is ivariat relative to T k ad sice T is ergodic Y is costat. M k k a.s. Y [0, ], (2.4) Lastly, let s uderstad the relatioship betwee L ad. We have see that give N k =, M k behaves like L. Now otice that for large k, N k = is approximately k 2, the average umber of poits i k k box, so k 2. Hece M k /k L /.

6 2-6 Lecture 2: Subadditive Ergodic Theorem Figure 2.4: A logest icreasig subsequece from a Poisso process. The solid lie shows the logest icreasig subsequece i Mk+m while the dotted lie shows the logest icreasig subsequece i M k ad Mm T k. Refereces [] R. Durrett. Probability: Theory ad Examples. Duxbury Press: Belmot, CA, 996. [2] G. Grimmett. Percolatio. Spriger-Verlag, 989. [3] J.M. Hammersley. A few seedligs of research. Proc. 6 th Berkeley Symp. Math. Stat. Prob., U. of Califoria Press, pages , 972. [4] J.F.C. Kigma. Subadditive ergodic theory. A. Probab., pages , 976. [5] J.M. Steele. Kigma s subadditive ergodic theorem. Aales de l Istitute Heri Poicare, pages 93 98, 989.