The definitions of a two-dimensional Markov shift and the associated Markov measure are given. Then a sufficient condition for the exponential mixing of such shifts is provided. This generalizes the well-known result in

the one-dimensional case

Md. Noorani, Mohd Salmi

Universiti Putra Malaysia Institutional Repository

Pertanika J. Sci. & Techno!. 4(1): 77-84 (1996)ISSN:0128-7680© Penerbit Universiti Pertanian MalaysiaThe Rate of Mixing of Two-dilllensionalMarkov ShiftsMohd. SalD1i Md. NooraniMathematics DepartmentFaculty of Mathematical ScienceUniversiti Kebangsaan Malaysia43600 UKM Bangi, Selangor D.E., MalaysiaReceived 28 May 1994ABSTRAKTakrif anjakan Markov matra dua berserta dengan sukatan Markov yangberpadanan diberi. Kemudian dipaparkan suatu syarat cukup untukpercampuran eksponen anjakan tersebut. Hasil ini mengitlakkan hasilyang diketahui umum dalam kasus matra satu.ABSTRACTThe definitions of a two-dimensional Markov shift and the associatedMarkov measure are given. Then a sufficient condition for the exponentialmixing of such shifts is provided. This generalizes the well-known result inthe one-dimensional case.Keywords: two-ditnensional Markov shifts, Markov measure, mixing rateINTRODUCTIONUnlike the theory of one-dimensional Markov shifts, the theory of higherdimensional analogue of these dynamical systems is filled with anomaliesand difficulties (see Schmidt 1990 for a briefsurvey). In this paper, we makea small contribution towards a better understanding of these higher-dimensional Markov shifts by studying the rate of mixing of such shifts.A standard result in the ergodic theory of one-dimensional Markovshifts is as follows: Let (T,X, C,f..t) be a (one-dimensional) Markov shiftwhere the Markov measure is given by some transition probability (p, P).Suppose A and B are arbitrary cylinder sets in C. Then the sequence(f..t(A n inB))n>O converges to f..t(A)f..t(B) at an exponential rate as n tendsto infinity, when the matrix P is irreducible and aperiodic (i.e., there existssome interger N> 0 such that all the entries of p N are strictly positive).This result follows from the crucial matrix fact that when P is irreducibleand aperiodic, then the sequence (pn(i,j))n>O converges exponentially fastto p(j) as n tends to infinity, for all i,j. Notethat an immediate corollary tothe above result is that T is strong-mixing.Mohd. Salmi Md. NooraniOur purpose in this short note is to generalize the aforementionedresults to the case of a two-dimensional Markov shift. Observe that thedynamical system in questions consists of two commuting (invertible)measure-preserving transformations acting on the measurable space offunctions from Z2 to some fixed finite set together with the Markovmeasure. Here the Markov measure is defined by two commuting stochasticmatrices P and Q such that they share a common stationary probabilityvector p (see below for details). Working analogously with the one-dimensional case, we need to look at the rate of convergence of the sequence(pmQn)(i,j)mn>O to p(j) as m,n tends to infinity, for all i,j. We show that ifeither P or Q. is irreducible and aperiodic then the convergence rate of theaforementioned sequence is exponentially fast. This in turn implies theexponential convergence of measures on rectangle sets for the correspondingtwo-dimensional Markov shift (see Theorem I). An immediate corollary tothe above is that the two-dimensional Markov shift is strong-mixing.DEFINITIONS AND RESULTSLet Ybe the finite set {1,2, ... ,k} equipped with the a-algebra 2Y . Themeasurable space ( yZ2 , B) is defined as the space of all functions x : Z2 ---t Yendowed with the product a-algebra B. Recall that this means, B is thesmallest a-algebra such that the collection of all projection maps1rF: yZ2 ---t yF which is given by 1rF(X) = XIF for each finite subset F ofZ2, is measurable. Of course, the set yF here is equipped with the producta-algebra fIe F 2Y. Given. x E yZ2, then we shall write Xc for the value off 2the function x at C E Z .We shall be interested in the following subsets of yZ2. First, let F be theset {C=(Cl,C2) EZ2 :at:::;ct:::;at+ ut,t=1,2} for some givena = (a1' a2) E Z2, U = (U1, U2) E (Z+)2. Then, an (elementary) rectangleRa,u is any subset of yZ2 which takes the formRa,u = {x E yZ2 : X(Cj,C2) = i(Cl,C2) , Vat:::; Ct :::; at + Ut, t = 1, 2}for some fixed elements ic of y, for each C E F. It is clear that such subsetsare measurable. Moreover, it is not difficult to see that the collection ofsuchrectangles generates the product a-algebra B.We shall now move on to the notion of a Markov measure on ( yZ2, B).For this, assume that we are given two k x k-matrices P and Q satisfying thefollowing three properties:I. P, Q are stochastic matrices such that PQ = QP.2. There exists a probability vector p = (P(I), ... ,p(k)) such thatpP = pandpQ = p.3. If pO, QO denotes the 0-1 matrices which are compatible with P and Qrespectively, then we require pOQo = QO pO and pOQo is also a 0-1matrix.78 Pertanika J. Sci. & Techno!. Vol. 4 No.1, 1996The Rate of Mixing of Two-dimensional Markov ShiftsLet{ Z2. }Ra,u = X E Y : X(CI,C2) = l(c\ ,C2)' Vat::; Ct ::; at + U/) t = 1,2be a rectangle, for some a = (ai, a2) E Z2 and U = (Ul, U2) E (Z+)2. We shallcall Ra,u an allowable rectangle if, in addition,pO (X(CI ,C2)' X(CI +1,c2)) = QO (X(CI ,C2)' X(CI ,C2+ 1)) = 1for all x E Ra,u and at ::; Ct ::; at + Ut - 1, t = 1,2. We are now ready todefine the Markov measure m on ( yZ 2, B) associated with the matrices Pand Q. Let Ra,u = {x E yZ2: X(Cl,C2) = i(ci ,C2)' Vat::; Ct ::; at + Ut, t = 1, 2} bean allowable rectangle. Then the measure of Ra,u is taken to beuI-lm(Ra,u) = p(i(al,a2)) II P(i(al+e,a2)' i(al+e+l,a2)) xe=OU2- 1II Q(i(a\+uI,a2+/), i(a\+uI,a2+1+ 1))1=0For non-allowable rectangles R, we take m(R) to be zero. By using theKolmogorov consistency theorem (see, e.g., Parthasarathy 1967), m extendsuniquely to a probability measure on the product (J-algebra B. In analogywith the one-dimensional case, we shall call this measure m the Markovmeasure defined by the matrices P and Q,We shall define the horizontal shift (J: yZ2 ---* yZ2 and the vertical shiftT: yZ2 ---* yZ2 by(crX)(CI,C2) = X(cI+l,c2) and (TX)(Cl,C2) = X(C\,C2+1)for all x E yZ2 and (CI, C2) E Z2. Then, it is clear that (J and T commute.Moreover, since each (J and T preserve the measure m on the algebra A offinite disjoint union of rectangles then they are measure-preserving on thesmallest (J-algebra containing A, which is, of course, the product (J-algebraB. Thus (J and T are two commuting measure-preserving automorphismsacting on ( yZ 2, B, m). We shall call the resulting (invertible) measure-preserving dynamical system ( yZ2, B, m, (J, T) a (two-dimensional) Markovshift with transition probability (p, P, Q).DISCUSSIONBy working on rectangle sets, it can be shown that the assumption thatpOQo is also a 0-1 matrix is needed to check consistency of the Markovmeasure (c.f. Kolmogorovs theorem).A second implication of the 0-1 assumption on the matrix pOQo is thatfor allowable rectangles Ra,u, m(Ra,u) is also given byPertanika J. Sci. & Techno!. Vo!. 4 No.1, 1996 79Mohd. Salmi Md. Nooraniu2- 1m(Ra,u) = p(i(al,a2)) IT P(i(al,a2+e') , i(a),a2+e'+I)) Xe'=0u)-lIT Q(i(al+f;,a2+u2)' ita) +1'+I,a2+u2))1'=0Suppose we give the set Y the discrete topology. Then the Markov measurem is supported by the subshift of finite typeX= {x E yZ2 : pO(X(e,f),X(e+1,f)) = QO(X(e,f),X(e,f+1)) = 1,Ve,fE Z}when we assume that the stationary probability vector p is a strictly positivevector. Note that the fact that X is non-empty follows from the commutingassumption on pO and QO.Using well-known methods from the theory of one-dimensional Markovshifts, we prove:Lemma 1Suppose P and Q. are two commuting k x k-stochastic matrices such that there existssome probability vector p = (P(1), ... ,p(k)) satisfyingpP = p = Po.: Ifeither P orQ. is irreducible and aperiodic then the sequence (pI! Q.I! (i, j) ) m, I!?-O converges to p(j)at an exponential rate as m, n tends to infiniry,for all i,j = 1,..., k, i.e.) there existsconstants C > 0) 0 :::; a, f3 :::; 1 with af3 < 1 such thatfor all m,n ~ 0and for all i,l = 1, ... ,k.ProofWithout loss on generality, we shall assume that P is irreducible andaperiodic. Thus by the Perron-Frobenius theorem (see, for instance, Seneta1981), the dominant eigenvalue 1 is a simple eigenvalue for P. Now let V bethe subspace {v E C k : <p,v >= O}. (Here <,> denotes the standardinner-product in C k ). Then, it is easy to see that P leaves V invariant, i.e.,PV C V. Moreover, if we denote the vector (1,1, ... ,1) by 1, then eachWEek can be uniquely written asW = W- < p, W > 1+ < p, w > 1such that w- < p, W > 1 E V and < p, W > 1 E U, where U is the one-dimensional subspace generated by the vector 1. Thus we haveC = VEfj U.Hence, by virtue of the simplicity of the eigenvalue 1 of P, we deduce that80 Pertanika J. Sci. & Techno!. Vo!. 4 No. I, 1996The Rate of Mixing of Two-dimensional Markov Shiftsthe spectral radius ofPlv (the restriction ofP to the invariant subspace V) isstrictly less than L The spectral radius formula then implies there existssome 0 :s: a < 1 such thatfor all m 2: 0 and some constant C1 2: O. (Here the matrix norm is the usualoperator norm.) Now, if Q also has 1 as a simple eigenvalue then using thesame argument as above, it gives usfor all n 2: 0 and some constants 0 :s: {3 < 1, C2 > O. Thus, in this case, wehavefor all m, n 2: 0 and some constant C > O.On the other hand, when the eigenvalue 1 of Q is no longer simple thespectral radius of Qlv may equal 1. In this case, it suffices to note that Qiv isbounded so thatfor all n 2: 0 and for some constant C3 > O. This in turn implies thatfor all m, n 2: 0, and some constant C' > O. Hence, in either case, we haveII~ Qiv II :s: Cam{3 n Vm,n 2: 0and for some constants C > 0,0 :s: a, {3 :s: 1 with a{3 < L From this lastinequality we deduce thatVm,n 2: 0, Vv E V.(Here the vector norm is the usual h norm.) Recall that given WE Ck , thenw- < p, W > 1 E V. Thus, since pmQn is stochastic for all m, n 2: 0, we havefor all m, n 2: 0 and all W E Ck . Furthermore, by takingW = (0, ... ,0,1,0, ... ,0), the j-th unit vector, it is easy to see thatfor all m, n 2: 0 and i,j = 1,2, ... , k. ThusVm,n 2: 0,Pertanika J. Sci. & Techno!. Vo!. 4 No. I, 1996 81Mohd. Salmi Md. Nooraniand for all i,j = 1,2, , k where C' = C max{llw- < P, w > 111: w = j-thunit vector, j = 1, 2, ,k}. This gives us the required result.Observe that the essential ingredient in the above proof is the fact thateither P or Q has a simple dominant eigenvalue 1 and the rest of thespectrum has a modulus strictly less than 1. Of course, a similar observationalso holds in the case of a one-dimensional Markov shift. The followingtheorem is the main result of this note.Theorem 1Let ( yZ2, B, m, cr, T) be a Markov shift with transition probability (p, P, Q).Suppose either P or Q is irreducible and aperiodic. Then given arry rectangle A, B E B,there exists an interger N > 0 and constants C> 0, 0 :S a, {3 :S 1 with a{3 < 1 suchthatfor all intergers m, n 2: N.ProofLet A, B E B be two arbitrary rectangles. Then, by definition, there existsa = (al,az),b = (b1,bz) E ZZ,U = (UI,UZ),V = (Vl,VZ) E (Z+)z such thatA = Ra,u and B = Rb,v whereandHenceAn cr-mT-nB = {x E yZ2: X(C\,C2) = i(c\,c2) , Vat :S CI :S al + UIand X(d1+m,d2+n) = i(dJ ,d2)' Vbt :S dt:S bt + VI' t = 1,2}Now, let m > al + UI - b l and n > az + Uz - bz. Then, in particular,An cr-mT-nB is a finite disjoint union of elementary rectanglesR], Rz, ... , Rk, say. If either A or B is non-allowable, then each of theR/s are non-allowable. Thus, in this case, we have m(A n cr-mT-nB) =~~] m(Ri ) = O. Moreover, since m(A)m(B) is also zero, then the requiredresult holds trivially in this case. We are now left with the case when theelementary rectangles A and B are both allowable. Then, by the assumptionpDQo is also a 0-1 matrix (see above), it is straightforward (but tedious) tocheck that8Z Pertanika J. Sci. & Techno!. Vo!. 4 No. I, 1996The Rate of Mixing of Two-dimensional Markov Shiftsul-Im(An(J"-mT-nB) = p(i(al,a2)) II P(i(al+e,a2) , i(al+e+l,a2)) Xe=Ou2-1II Q(i(al+ul,a2+f) , i(al+Ul,a2+f+I))pm' Qn' (i(al+ul,a2+u2) , i(bl,b2)) X1=0vl-I V2- 1II P(i(bl+e,b2)' i(b1+e+l,b2)) II Q(i(b1+Vl,b2+ll' i(b1+VI,b2+f+I))e=O 1=0where m' = bl + m - (al +UI) > 0, n' = b2 + n - (a2 + U2) > O. Observethat since one of P or Q is aperiodic, then the stationary probability vector pis strictly positive. HenceSo thatThus, by combining the previous line and Lemma 1, we gather that there•exists 0 ::::: a, (3 ::::: 1 with a(3 < 1 such thatIm(A n (J"-mT-nB) - m(A)m(B)1 ::::: C am'(3n'for all m', n' > 0 and for some constant C> O. Finally, by takingN = max((al + UI) - bl , (a2 + U2) - b2, 1), we deduce thatIm(A n (J"-mT-nB) - m(A)m(B)1 ::::: C' am(3nfor all m, n ~ N and some constant C' > O. This then gives us the requiredresult. DLet Tj, T2 be two commuting measure-preserving transformationsacting on the probability space (2, D, v). Then the resulting dynamicalsystem is said to be strong-mixing iflim m(A n TJmTinB) = m(A)m(B)m,n---toofor all A,B E D.Recall that for Markov shifts, disjoint unions of rectangles form analgebra that generates the product (J"-algebra. Hence, by using a standardapproximation theorem (see, e.g., Walters 1981), we have an immediatecorollary to Theorem 1.Pertanika J. Sci. & Techno!. Vo!. 4 No. I, 1996 83Mohd. Salmi Md. NooraniCorollary 1Let ( yZ2, B, m, a, T) be a Markov shift with transition probability (p, P, Q). Ifeither P or Q is irreducible and aperiodic then the Markov shift is strong-mixing.Observe that if P, say, is irreducible and aperiodic and Q is the identitymatrix then we can identify the two-dimensional Markov shift with the one-dimensional Markov shift with transition probability (p, P). Thus we canretrieve the well-known mixing result for one-dimensional Markov shiftsfrom the above corollary. As a final remark, note that all the above resultscan be generalized for Markov shifts of arbitrary dimensions.ACKNOWLEDGEMENTSWe would like to express our gratitude to Prof W. Parry for helpfuldiscussions through-out the course of this work. We also would like thankthe referee for the helpful suggestions to improve the presentation of thispaper.REFERENCESPARTHASARATHY, K. L. 1967. Probability Measures on Metric Spaces. New York:Academic Press.SCHMIDT, K. 1990. Algebraic Ideas in Ergodic Theory. CBMS Regional c.onf. Ser. inMath., No. 76. Providence RI: Amer. Math. Soc.SENETA, E. 1981. Non-negative Matrices and Markov Chains, 2nd edn. New York:Springer-Verlag.WALTERS, P. 1981. An Introduction to Ergodic Theory, G.T.M. 79. Berlin: Springer-Verlag.84 Pertanika J. Sci. & Techno!. Vo!. 4 No. I, 1996

The Rate of Mixing of Two-dimensional Markov Shifts

http://psasir.upm.edu.my/3277/1/The_Rate_of_Mixing_of_Two-dilllensional.pdf

The Rate of Mixing of Two-dimensional Markov Shifts

Abstract

Similar works

Full text

Available Versions

Universiti Putra Malaysia Institutional Repository