Cs aki and Vincze have de fined in 1961 a discrete transformation T which
applies to simple random walks and is measure preserving. In this paper, we are
interested in ergodic and assymptotic properties of T . We prove that T is
exact : \cap_{k\geq 1} \sigma(T^k(S)) is trivial for each simple random walk S
and give a precise description of the lost information at each step k. We then
show that, in a suitable scaling limit, all iterations of T "converge" to the
corresponding iterations of the continous L evy transform of Brownian motion.
Some consequences are also derived from these two results.Comment: Title changed and various other modification

Hajri, Hatem

Studia Scientiarum Mathematicarum Hungarica

English

arXiv

Csáki and Vincze have defined in 1961 a discrete transformation T which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and asymptotic properties of T. We prove that T is exact: ∩k≧1σ(Tk(S)) is trivial for each simple random walk S and give a precise description of the lost information at each step k. We then show that, in a suitable scaling limit, all iterations of T “converge” to the corresponding iterations of the continuous Lévy transform of Brownian motion.</jats:p

Hatem Hajri

Crossref

On the csáki-vincze transformation

Open Repository and Bibliography - Luxembourg

Studia Scientiarum Mathematicarum HungaricaDOI: 10.1556/SScMath.2013.11240ON THE CSAKI{VINCZE TRANSFORMATIONHATEM HAJRIUniversite du LuxembourgAuthor: please provide full mailing address.e-mail: Hatem.Hajri@uni.luCommunicated by E. Csaki(Received November 8, 2012; accepted March 18, 2013)AbstractCsaki and Vincze have dened in 1961 a discrete transformation T which applies tosimple random walks and is measure preserving. In this paper, we are interested in ergodicand asymptotic properties of T . We prove that T is exact: Tk=1 (T k(S)) is trivial foreach simple random walk S and give a precise description of the lost information at eachstep k. We then show that, in a suitable scaling limit, all iterations of T \converge" to thecorresponding iterations of the continuous Levy transform of Brownian motion.1. Introduction and main resultsLet B be a Brownian motion, then T (B)t =R t0 sgn(Bs)dBs is a Brownianmotion too. Iterating T yields a family of Brownian motions (Bn)n given byB0 = B; Bn+1 = T (Bn):We call Bn the n-iterated Levy transform of B. At least two transforma-tions of simple random walks have been studied in the literature as discreteanalogues to T . For a simple random walk (SRW) S, Dubins and Smorodin-sky [2] dene the Levy transform  (S) of S as a SRW obtained from thepathsn 7 ! jSnj   Ln2000 Mathematics Subject Classication. Author: please provide MSC classication.Key words and phrases. Author: please provide keywords.c 2013 Akademiai Kiado, Budapest2 H. HAJRIwhere L is a discrete analogue of local time. Their fundamental result saysthat S can be recovered from the signs of the excursions of S; (S); 2(S); : : :and a fortiori   is ergodic. Later, another discrete Levy transformation Fwas given by Fujita [4]:F (S)k+1 F (S)k = sgn(Sk)(Sk+1 Sk); with the convention sgn(0) =  1:However, F is not ergodic by the main result of [4]. Our main purpose in thispaper is to study a transformation already obtained by Csaki and Vincze.Let W = C(R+;R) be the Wiener space equipped with the distancedU (w;w0) =Xn=12 n sup05t5njw(t)  w0(t)j ^ 1 :We endow E =WN with the product metric dened for each x = (xk)k=0,y = (yk)k=0 byd(x; y) =Xk=02 k(dU (xk; yk) ^ 1):Thus (E; d) is a separable complete metric space.For each SRW S and h = 0, we denote by T h(S) the h-iterated Csaki{Vincze transformation (to be dened in Section 2.1) of S with the conventionT 0(S) = S.Let B be a Brownian motion dened on (;A;P). For each n = 1, deneTn0 = 0 and for all k = 0,Tnk+1 = inft = Tnk : jBt  BTnk j =1pn:Then Snk =pnBTnk , k = 0, is a SRW and we have the followingTheorem 1.(i) For each SRW S and h = 0, T h(S) is independent of (Sj ; j 5 h) anda fortiori \h=0(T h(S))is trivial.(ii) For each n = 1, h = 0 and t = 0, deneSn;h(t) =1pnT h(Sn)bntc +(nt  bntc)pn T h(Sn)bntc+1   T h(Sn)bntc :ON THE CSAKI{VINCZE TRANSFORMATION 3Then(Sn;0; Sn;1; Sn;2; : : : )converges to(B0; B1; B2; : : : )in probability in E as n!1.Theorem 1(i) says that T is exact, but there are more informations in theproof. For instance, the random vectors (S1; S2; : : : ; Sn) and (S1;T (S)1; : : : ;T n 1(S)1) generate the same -eld; so the whole path (Sn)n=1 can be en-coded in the sequence (T n(S)1)n=0 which is stronger than exactness. FromTheorem 1(i), we can deduce the followingCorollary 1. Fix p = 2 and let i = (in)n=1, i 2 [1; p] be p nonnegativesequences such that1n  ! +1; in   i 1n  ! +1 as n!1 for all i 2 [2; p]:Let S be a SRW and W1; : : : ;Wp+1 be p+ 1 independent Brownian motions.For n = 1; h = 0; t 2 R+, dene(1) Shn(t) =1pnT h(S)bntc +(nt  bntc)pn T h(S)bntc+1   T h(S)bntc :Then S0n; Sbn1ncn ; : : : ; Sbnpncnlaw     !n!+1 W1;W2; : : : ;Wp+1in Wp+1:A natural question which is actually motivated by the famous question ofergodicity of the Levy transformation T as it will be discussed in Section 3,is to focus on sequences (hn)n tending to 1 and satisfying(2) limn!1 (Sn;hn(t) Bhnt ) = 0 in probability:Such sequences exist and when (2) holds, we necessarily have limn!1 hnn =0. This is summarized in the followingProposition 1. With the same notations of Theorem 1 :(i) There exists a family (i)i2N of nondecreasing sequences i = (in)n2Nwith values in N such that0n  ! +1; in   i 1n  ! +1 as n!1 for all i = 14 H. HAJRIand moreoverlimn!1 (Sn;in  Bin) = 0 in probability in Wfor all i 2 N.(ii) If (hn)n is any integer-valued sequence such thathnn does not tend to 0,then there exists no t > 0 such that (2) holds.In the next section, we review the Csaki{Vincze transformation, establishpart (i) of Theorem 1 and show that (S; T (S); : : : ; T h(S); : : : ) \converges"in law to (B;T (B); : : : ; T h(B); : : : ). To prove part (ii) of Theorem 1, we usethe simple idea: if Zn converges in law to a constant c, then the convergenceholds also in probability. The other proofs are based on the crucial propertyof the transformation T : T h(S) is independent of (Sj ; j 5 h) for each h.In Section 3, we compare our work with [2] and [4] and discuss the famousquestion of ergodicity of T .2. Proofs2.1. The Csaki{Vincze transformation and convergence in lawFor the sequel, we recommend to read the pages 113 and 114 in [7] (The-orem 2 below). Some consequences (see Proposition 2 below) have beendrawn in [5] (Sections 2.1 and 2.2). We also notice that our stating of thisresult is slightly dierent from [7] and leave to the reader to make the obviousanalogy.Theorem 2 ([7] page 113). Let S = (Sn)n=0 be a SRW dened on(;A;P) and Xi = Si   Si 1, i = 1. Dene 0 = 0 and for l = 0,l+1 = mini > l : Si 1Si+1 < 0	:SetXj =Xl=0( 1)lX1Xj+11fl+15j5l+1g:Then S0 = 0, Sn = X1+   +Xn, n = 1 is a SRW. Moreover if Yn := Sn mink5nSk, then for all n 2 N,(3) jYn   jSnjj 5 2:We call S = T (S), the Csaki{Vincze transformation of S (see the gures 1and 2 below).ON THE CSAKI{VINCZE TRANSFORMATION 5Fig. 1S and T (S)Fig. 2jSj and YNote that ( 1)lX1 is simply equal to sgn(S)j[l+1;l+1](:= Xl+1) whichcan easily be checked by induction on l. Thus for all j 2 [l + 1; l+1],Xj = sgn(S)j[l+1;l+1](Sj+1   Sj)or equivalently(4) T (S)j   T (S)j 1 = sgn(Sj  12)(Sj+1   Sj)6 H. HAJRIwhere t  ! St is the linear interpolation of (Sn)n=0. Hence, one can expectthat (S; T (S)) will \converge" to (B;B1) in a suitable sense. The followingproposition has been established in [5]. We give its proof for completeness.Proposition 2. With the same notations of Theorem 2, we have(i) For all n = 0, (T (S)j ; j 5 n) _ (S1) = (Sj ; j 5 n+ 1).(ii) S1 is independent of (T (S)).Proof. (i) The inclusion  is clear from (4). Now, for all 1 5 j 5 n,Xj+1 =Xl=0( 1)lX1Xj1fl+15j5l+1g. As a consequence of (iii) and (iv) [7](page 114), for all l = 0,l = min fn = 0; T (S)n =  2lg:Thus l is a stopping time with respect to the natural ltration of T (S)and as a result fl + 1 5 j 5 l+1g 2 (T (S)h; h 5 j   1) which proves theinclusion .(ii) We may write for all l = 1,l = min fi > l 1 : X1Si 1X1Si+1 < 0g:This shows that T (S) is (X1Xj+1; j = 0)-measurable and (ii) is proved. Note thatT (S) = T ( S); (T h+1(S))  (T h(S));which is the analogue ofT (B) = T ( B); (T h+1(B))  (T h(B)):The previous proposition yields the followingCorollary 2. For all n = 0,(i) (S) = (T n(S)) _ (Sk; k 5 n).(ii) (T n(S)) and (Sk; k 5 n) are independent.(iii) The -eldG1 =\n=0(T n(S))is P-trivial.ON THE CSAKI{VINCZE TRANSFORMATION 7Proof. Set Xi = Si   Si 1, i = 1.(i) We apply successively Proposition 2(i) so that for all n = 1,(S) = (T (S)) _ (S1)= (T 2(S)) _ (T (S)1) _ (S1)=   = (T n(S)) _ (T n 1(S)1) _    _ (T (S)1) _ (S1):To deduce (i), it suces to prove that(5) (Sk; k 5 n) = (T n 1(S)1) _    _ (T (S)1) _ (S1):Again Proposition 2(i), yields(Sk; k 5 n) = (T (S)j ; j 5 n  1) _ (S1)= (T 2(S)j ; j 5 n  2) _ (T (S)1) _ (S1)=   = (T n 1(S)1) _ (T n 2(S)1)    _ (T (S)1) _ (S1)which proves (5) and allows to deduce (i).(ii) will be proved by induction on n. For n = 0, this is clear. Supposethe result holds for n, then S1; T 1(S)1; : : : ; T n 1(S)1; T n(S) are indepen-dent (recall (5)). Let us prove that S1; T 1(S)1; : : : ; T n(S)1; T n+1(S) areindependent which will imply (ii) by (5). Note that T n(S)1 and T n+1(S)are (T n(S))-measurable. By the induction hypothesis, this shows that(S1; T 1(S)1; : : : ; T n 1(S)1) and (T n(S)1; T n+1(S)) are independent. ButT n(S)1 and T n+1(S) are also independent by Proposition 2(ii). Hence (ii)holds for n+ 1 and thus for all n.(iii) Let A 2 G1 and x n = 1. Then A 2 (T n(S)) and we deduce from(ii) that A is independent of (Sk; k 5 n). Since this holds for all n, A isindependent of (S). As G1  (S), A is therefore independent of itself. Let S be a SRW dened on (;A;P) and recall the denition of Shn(t)from (1). On E, deneZn(t0; t1; : : : ; th; : : : ) =S0n(t0); S1n(t1);    ; Shn(th);   and let Pn be the law of Zn.Lemma 1. The family fPn; n = 1g is tight on E.8 H. HAJRIProof. By Donsker theorem for each h, Shn converges in law to standardBrownian motion as n!1. Thus the law of each coordinate of Zn is tighton W which is sucient to get the result (see [3] page 107). The limit process. Fix a sequence (mn; n 2 N) such that Zmn law     !n!+1Z in E whereZ = (B(0); B(1); : : : ; B(h); : : : )is the limit process. Note that B(0) is a Brownian motion. From (3), wehave 8n = 1, t = 0jS0n(t)j   (S1n(t)  min05u5tS1n(u)) 5 2012pn :Letting n!1, we getjB(0)t j = B(1)t   min05u5tB(1)u :Tanaka's formula for local time givesjB(0)t j =tZ0sgn(B(0)u )dB(0)u + Lt(B(0)) = B(1)t   min05u5tB(1)u ;where Lt(B(0)) is the local time at 0 of B(0) and soB(1)t =tZ0sgn(B(0)s )dB(0)s :The same reasoning shows that for all h = 1,B(h+1)t =tZ0sgn(B(h)s )dB(h)s :Thus the law of Z is independent of the sequence (mn; n 2 N) and therefore(6) (S0n; S1n; : : : ; Shn; : : : )law     !n!+1 (B;B1; : : : ; Bh; : : : ) in Ewhere B is a Brownian motion.ON THE CSAKI{VINCZE TRANSFORMATION 92.2. Convergence in probabilityLet B be a Brownian motion and recall the notations in Theorem 1. Foreach n = 1, deneUn = (B;Sn;0; B1; Sn;1; : : : ; Bh; Sn;h; : : : ):and let Qn be the law of Un. Since T h(Sn) is a simple random walk for each(h;n), a similar argument as in the proof of Lemma 1 shows that fQn; n = 1gis tight on E. Fix a sequence (mn; n 2 N) such that Umn law     !n!+1 U in E.Using (6), we see that there exist two Brownian motions C and D such thatU = (C;D;C1; D1; : : : ; Ch; Dh; : : : ):It is easy to check that if ' :W  ! R is bounded and uniformly continuous,then 	(f; g) = '(f   g) dened for all (f; g) 2W2 is also bounded uniformlycontinuous which comes fromdU (f   f 0; g   g0) = dU (f   g; f 0   g0) for all f; f 0; g; g0 2W:Thus if (Fn; Gn) converges in law to (F;G) in W2, then Fn  Gn convergesin law to F  G in W. Applying this, we see that B   Sn;0 converges in lawto C  D. On the other hand, B   Sn;0 converges to 0 (in W) in probability(see [6] page 39). Consequently C = D andUnlaw     !n!+1 (B;B;B1; B1; : : : ; Bh; Bh; : : : ) in E:In particular for each h, Sn;h  Bh converges in law to 0 as n!1, that isSn;h converges to Bh in probability as n!1. Now the following equiva-lences are easy(i) limn!1(Sn;0; Sn;1; Sn;2; : : : ) = (B0; B1; B2; : : : ) in probability in E.(ii) For each h, limn!1 Sn;h = Bh in probability in W.Since we have proved (ii), Theorem 1 holds.2.3. Proof of Corollary 1(i) Let S be a SRW and W1;W2; : : : ;Wp+1 be p+ 1 independent Brow-nian motions (not necessarily dened on the same probability space as S).Fix0 5 t01 5    5 t0i0 ; 0 5 t11 5    5 t1i1 ; : : : ; 0 5 tp1 5    5 tpip :10 H. HAJRIBy Corollary 2(ii), for n large enough (such that bnt0i0c+1 5 bn1nc), S0n(t01);: : : ; S0n(t0i0)which is (Sj ; j 5 bnt0i0c+ 1)-measurable, is independent ofT bn1nc(S). Thus (S0n(t01); : : : ; S0n(t0i0)) is independent of (Sbn1ncn ; : : : ; Sbnpncn )and similarly T bn2nc(S) is independent of (T bn1nc(S)j , j 5 bn2nc bn1nc).Again, for n large (such that bnt1i1c+ 1 5 bn2nc   bn1nc), Sbn1ncn (t11); : : : ;Sbn1ncn (t1i1)is (T bn1nc(S)j ; j 5 bn2nc   bn1nc)-measurable and thereforeis independent of Sbn2ncn ; : : : ; Sbnpncn. By induction on p, for n largeenough,  S0n(t01); : : : ; S0n(t0i0);Sbn1ncn (t11); : : : ; Sbn1ncn (t1i1); : : : ;Sbnpncn (tp1); : : : ; Sbnpncn (tpip)are independent and this yields the convergence in law ofS0n(t01); : : : ; S0n(t0i0); Sbn1ncn (t11); : : : ; Sbn1ncn (t1i1); : : : ;Sbnpncn (tp1); : : : ; Sbnpncn (tpip)to(W1(t01); : : : ;W1(t0i0);W2(t11); : : : ;W2(t1i1); : : : ;Wp+1(tp1); : : : ;Wp+1(tpip)):Thus the convergence of the nite dimensional marginals holds and the proofis completed.2.4. Proof of Proposition 1To prove part (i), we need the following lemma which may be found in [1]page 32 in more generality:Lemma 2. If (uk;n)k;n2N is a nonnegative and bounded doubly indexedsequence such that for all k; limn!1 uk;n = 0, then there exists a nondecreas-ing sequence (kn)n such that limn!1 kn = +1 and limn!1 ukn;n = 0.Proof. By induction on p, we construct an increasing sequence (np)p2Nsuch that up;n < 2 p for all n = np. Now denekn =(n if 0 5 n 5 n0p if np 5 n < np+1 for some p 2 N:ON THE CSAKI{VINCZE TRANSFORMATION 11Clearly n 7 ! kn is nondecreasing and limn!1 kn = +1. Moreover for all pand n = np, we have ukn;n < 2 p. Thus for all p, 0 5 lim supn!1 ukn;n 5 2 pand since p is arbitrary, the lemma is proved. The previous lemma applied touk;n = E[dU (Sn;k; Bk) ^ 1];guarantees the existence of a nondecreasing sequence (0n)n with values in Nsuch that limn!1 0n = +1 and(7) limn!1 Sn;0n  B0n = 0 in probability in W:Now setV n = (B0n ; Sn;0n ; B0n+1; Sn;0n+1; : : : ; B0n+h; Sn;0n+h; : : : ):Using the same idea as in Section 2.2 and the relation (7), we prove that forall j 2 N,(8) limn!1 Sn;0n+j  B0n+j = 0 in probability in W:Equivalently: for all j 2 N,limn!1u0j;n = 0 where u0j;n = EdU (Sn;0n+j ; B0n+j) ^ 1 :By Lemma 2 again, there exists a nondecreasing sequence (0n)n with valuesin N such that limn!1 0n = +1 and(9) limn!1 Sn;0n+0n  B0n+0n = 0 in probability in W:Dene 1n = 0n + 0n. Now using (9) and the same preceding idea, we con-struct 2 and all the (i)i by the same way. Thus part (i) of Proposition 1is proved.To prove (ii), writeT (Sn)j+1   T (Sn)j = sgn(Snj+ 12)(Snj+2   Snj+1):12 H. HAJRIThus for each k = 1 and i = 1,T k(Sn)i =i 1Xj=0Pn;k;j(Snj+k+1   Snj+k); withPn;k;j =kYl=1sgnT k l(Sn)j+l  12:Denote by (Ft)t=0 the natural ltration of B, then Pn;k;j is the product ofk random signs which are FTnj+k -measurable. This yieldsE[Pn;k;j(Snj+k+1   Snj+k)jFTnj+k]= Pn;k;jE[pn(BTnj+k+1  BTnj+k)jFTnj+k]= 0:Consequently E[T k(Sn)ijFTnj+k ] = 0 and a fortioriE[Sn;k(t)jFTnk ] = 0 for all n; k and t:Suppose there exists t > 0 (which is xed from now) such thatlimn!1 (Sn;hn(t) Bhnt ) = 0in probability; we will show that hnn must tend to 0. By Burkholder's in-equality, we haveE[jSj jp] 5 CpEh S21 + (S2   S1)2 +   + (Sj   Sj 1)2 p2i= Cpjp2 :Hence the Lp-norm of Sn;hn(t) is bounded uniformly in n and the same istrue for the Lp-norm of Bhnt because Bhn is a Brownian motion. As a con-sequence, Sn;hn(t) Bhnt tends to 0 also in Lp-spaces.From ESn;hn(t)jFTnhn= 0 and using the L2-continuity of conditionalexpectations, we getEBhnt jFTnhn ! 0 in L2:Since Ms = Bhnt^s is a square-integrable F-martingale; we have E[Bhnt jFTnhn ]= Bhnt^Tnhnand Bhnt^Tnhnmust therefore tend to 0 in L2. So0 = limn!1Eh(Bhnt^Tnhn)2i= limn!1E[t ^ Tnhn]:ON THE CSAKI{VINCZE TRANSFORMATION 13This means that Tnhn ! 0 in probability. Now recall the followingLemma 3 (see [6] page 39). The sequence of continuous-time processes(n)n dened byn(t) =knfor t 2 Tnk ; Tnk+1converges uniformly on compacts in probability to the identity process t.This Lemma implies that n(Tnhn)! 0 in probability. But n(Tnhn) =hnn , so thathnn ! 0.3. Concluding remarksWe rst notice that with a little more work, Theorem 1(ii) remains truewhen the Csaki{Vincze transform is replaced with the Dubins{Smorodinsky,Fujita transform or any other\reasonable"discrete version. In Proposition 1,it is clear that there is no contradiction between the two statements. In fact,by the proof of Lemma 2, the sequences (i)i2N are constructed such that0 5 0n 5 n and 0 5 in   i 1n 5 n for all i and n. Let us now explain ourinterest in relation (2). Suppose that (2) were true for a sequence (hn)n withhnn !1, then using the convergence of Sn;0 to B and Corollary 2(ii) appliedto Sn, this would imply that (B;Bhn) converge in law to a 2-dimensionalBrownian motion which is equivalent to the ergodicity of the continuousLevy transformation on path space (see Proposition 17 in [8]). Corollary 1asserts that this convergence holds in discrete time. However as proved be-fore, such a sequence (hn)n does not exist and so the possible ergodicity of Tcannot be established by arguments involving assymptotics of T n. Thus theimpression that a thorough study of good discrete versions could lead to abetter understanding of the conjectured ergodicity of T may be misleading.REFERENCES[1] Attouch, H., Variational convergence for functions and operators, ApplicableMathematics Series, Pitman (Advanced Publishing Program), Boston, MA,1984.[2] Dubins, Lester E. and Smorodinsky, Meir, The modied, discrete, Levy-trans-formation is Bernoulli, in: Seminaire de Probabilites, XXVI, volume 1526 ofLecture Notes in Math., pages 157{161. Springer, Berlin, 1992.[3] Ethier, S. and Kurtz, T., Markov processes: Characterization and convergence.Wiley Series in Probability and Mathematical Statistics. 1986.[4] Fujita, Takahiko, A random walk analogue of Levy's theorem. Studia Sci. Math.Hungar., 45(2) (2008), 223{233.14 H. HAJRI: ON THE CSAKI{VINCZE TRANSFORMATION[5] Hajri, Hatem, Discrete approximations to solution ows of Tanaka's SDE relatedto Walsh Brownian motion, in: Seminaire de Probabilites XLIV, volume 2046of Lecture Notes in Math., pages 167{190. Springer, Heidelberg, 2012.[6] Ito^, Kiyosi and McKean, Henry P., Jr., Diusion processes and their samplepaths, Second printing, corrected, Die Grundlehren der mathematischen Wis-senschaften, Band 125. Springer-Verlag, Berlin{New York, 1974.[7] Revesz, P., Random walk in random and non-random environments. Second edi-tion, World Scientic Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.[8] Prokaj, Vilmos, Some sucient conditions for the ergodicity of the Levy transfor-mation, to appear in Seminaire de Probabilites, available at http://arxiv.org/pdf/1206.2485.pdf, 2012.

On the Csaki Vincze transformation

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