Reconstructing a random scenery in polynomial time

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1 Reconstructing a random scenery in polynomial time Citation for published version (AA): Matzinger, H., & Rolles, S. W. W. (). Reconstructing a random scenery in polynomial time. (Report Eurandom Vol. 1). Eindhoven: Technische Universiteit Eindhoven. Document status and date: ublished: 1/1/ Document Version: ublisher s DF, also known as Version of Record (includes final page, issue and volume numbers) lease check the document version of this publication: A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. eople interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 5fa of the Dutch Copyright Act, indicated by the Taverne license above, please follow below link for the End User Agreement: Take down policy If you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim. Download date: 11. Mar. 19

2 Reconstructing a random scenery in polynomial time Heinrich Matzinger Silke W.W. Rolles y March 1, Abstract Benjamini asked whether the scenery reconstruction problem can be solved in polynomial time. In this article, we answer his question in the armative for an i.i.d. uniformly colored scenery on Z observed along a random walk path with bounded jumps. We assume the random walk is recurrent, can reach every integer with positive probability, and the number of possible single steps for the random walk exceeds the number of colors. We prove that a nite piece of scenery of length l around the origin can be reconstructed up to reection and a small translation from the rst p(l) observations with high probability here p is a polynomial and the probability that the reconstruction succeeds converges to 1 as l! Introduction and Result We call a coloring of Z with colors from the nite set C := f1 ::: Cg a scenery. The scenery reconstruction problem can be described as follows: Let be a scenery and let S := (S k k N ) be a recurrent random walk on Z. If we are given the color record := ((S k ) k N ) observed along the random walk path, can we almost surely reconstruct from these observations (of course without knowing S)? It is not hard to see that in general the reconstruction works only up to a reection and translation. Recently, Lindenstrauss [7] proved the existence of uncountably many sceneries which cannot be reconstructed. Nevertheless it turns out that in many situations typical" sceneries can be reconstructed if the scenery is randomly colored and the random walk S is independent of. Work on the scenery reconstruction problem started with the question how much information can be extracted from the color record. This question was Eurandom, O Box 513, 5 MB Eindhoven, The Netherlands. Matzinger@eurandom.tue.nl y Eurandom, O Box 513, 5 MB Eindhoven, The Netherlands. Rolles@eurandom.tue.nl 1 Keywords: Scenery reconstruction, random walk, polynomial time 1

3 addressed in the 8s by Benjamini and independently by den Hollander and Keane (see []). Kesten [5] proved that a single defect in a 4-color random scenery can be detected if the scenery is i.i.d. uniformly colored. In his h.d. thesis [11] (see also [1] and [13]) Matzinger studied the scenery reconstruction problem for i.i.d. uniformly colored -color sceneries. He showed that almost all sceneries can be almost surely reconstructed up to equivalence if they are observed along a simple random walk path (with holding) here we call two sceneries and equivalent,, if can be obtained from by a reection and/or a translation. After Kesten had noticed that Matzinger's proof relies heavily on the skip-freeness of the random walk, Lowe, Matzinger, and Merkl [1]proved that scenery reconstruction still works if the random walk has i.i.d. increments with nite support. More precisely they made the following assumptions: The scenery on Z is i.i.d. uniformly colored with colors from the set C. The random walk is independent of the scenery, has i.i.d. increments with nite support, is recurrent, and can reach every integer with positive probability. They require that there is at least one color more than possible jumps for the random walk. Under these assumptions it is proved in [1] that almost all sceneries can be almost surely reconstructed up to equivalence. In this article, we rene the reconstruction result of Lowe, Matzinger, and Merkl. We make the same assumptions on scenery and random walk as in [1] with the only exception that we assume that the maximal jump lengths to the left and to the right of the random walk are equal. This additional assumption is made to keep the exposition as easy as possible. We believe that our result is true without this assumption. Below weprove that for n large a nite piece of scenery of length l(n) =1 n + 1 around the origin can be reconstructed with high probability from the rst n 7 + 1n observations with a constant > thus the number of observations needed is polynomial in l(n). In order to state our main result we need some notation. All intervals are taken over the integers, e.g. [a b] :=fx Z : a x bg. We write fjd for the restriction of a function f to a set D. For two words w C [ m] and w C [ m ] with m m we write w 4 w if there exists an interval I [ m ] such that the restriction of w to I can be obtained from w by a reection and/or a translation. We set for n N n 1 := bp n c n := bp n 1c (1) here bxc denotes the largest integer x. The dependence of n on n is always suppressed wealways write n instead of using the more precise notation n (n ). Formally, our result reads as follows: N there Theorem 1 There exists > such that for innitely many n exists a map A n : [ C[ n +n7 +1n!C [5n 5 n ] which is measurable with respect to the -algebras generated by the canonical projections such that for the events j[ n E n j [1 n 1 n ] ini works := j n1 n1 4 A n + n 7 + 1n [ 4

4 the following holds: lim n!1 (En ini works )=1: The algorithm A n gets as input the rst n +n 7 + 1n observations of the random walker and produces an output of length 1 n +1. If the reconstruction is successful in the sense that the event E n ini works holds, then the output of A n is a piece of the scenery which istypically not centered around the origin. The output is a piece of scenery which is up to equivalence contained in [1 n 1 n ]. Note that it is essential to reconstruct a nite piece of scenery close to the origin because the scenery is i.i.d. uniformly colored and therefore any nite color pattern can be almost surely found somewhere in the scenery. In [14], the authors study the scenery reconstruction problem under the same assumptions as in this article and prove that almost all sceneries can be almost surely reconstructed up to equivalence if there are some errors in the observations. Scenery reconstruction results in dierent settings have e.g. been obtained by Lowe andmatzinger in [9] and [8]. Related work on the scenery distinguishing problem has been done by Benjamini and Kesten [1] and Howard in [4] and [3]. The remainder of this article is organized as follows: In Section, we formally dene our setting. In Section 3, we review a result from [1]. Section 4 contains the denition of the reconstruction algorithm A n. Section 5 contains some lemmas. In Section, we prove Theorem 1. Setting Let C. We assume that the scenery := ( k k Z) isi.i.d. uniformly distributed over C := f1 ::: Cg. Let be a probability measure over Z with nite support M and let S := (S k k N ) be a random walk on Z with -distributed increments, independent of. We assume jmj < jcj, i.e. there is at least one color more than possible steps for the random walk. Furthermore we assume that M has greatest common divisor 1 and km k(k) = thus S can reach every integer with positive probability and is recurrent. To keep the notation simple we require max M = j min Mj, i.e. the maximal jump lengths to the left and to the right agree. We denote the maximal jump length by L. Let Z N be the set of all paths with jump sizes S k+1 S k Mfor all k N. We realize ( S) as canonical projections on the probability space := C Z := Z Q with the product -algebra generated by the canonical projections here denotes the uniform distribution on the set of colors C and Q the distribution of the random walk starting at the origin. We write := ( k := (S k ) k N ) for the observations of the scenery along the random walk path. 3

5 We dene the shift :! by ( S) 7! (( + S(1)) S( +1) S(1)) : Intuitively, shifts the origin to the position of the random walker at time 1. All constants keep there meaning throughout the article. 3 Review of a Result of Lowe, Matzinger, and Merkl Lowe, Matzinger, and Merkl [1] showed the existence of measurable maps A n which do partial reconstructions". In order to dene the reconstruction algorithm A n,we use these partial algorithms. The maps An reconstruct with high probability a large piece of scenery around the origin if the observations start with a suciently large block ofones. More formally: We dene the event thata n reconstructs correctly a piece of scenery around the origin: E n recon := j[ n n ] 4 A n j[ 1n [ 4 j[9 n 9 n ] here = (jcj ) is a suciently large positive constant independent of n. For n N, we denote by E B (n) theevent that the rst n + 1 observations are all equal to 1: E B (n) :=f k = 1 for all k [ n]g: For an interval J Z, we write (1) J for the piece of scenery in C J which is constantly equal to 1. We dene the event that there is a long block ofones close to the origin": BigBlock := There exists an integer interval J Ln Ln with jj jn 4 such thatjj =(1) J We denote by B the image of the conditional distribution (je B (n )) under the shift n, and we abbreviate ~ := B (jbigblock). We set " 1 (n ):= B ([E n recon] c ) () i.e. " 1 (n ) is the probability thata n does not fulll its task, and we observe B ([E n recon] c ) B ([E n recon] c BigBlock)+ B ([BigBlock] c ) ~ ([E n recon] c )+ B ([BigBlock] c ) : (3) In [1], Lowe, Matzinger, and Merkl prove lim n!1 ~ ([Erecon] n c ) =. The second term on the right-hand side of (3) converges to as n!1by Lemma 3.3 in [1]. Hence it follows from () and (3) that lim " 1(n )=: (4) n!1 : 4

6 4 The Algorithm A n In general it will not take too long" until a long block ofonesisobserved in the observations we then apply A n to the observations collected right after this long block ofones. Formally we let for k N, Z k be the Bernoulli random variable taking values in f 1g which is equal to one if and only if (kn + j) = 1 for all j [ n ]. We set k := min fk :Z k =1g : (5) +n7 +1n [! C [5n 5 n ] as fol- Denition We dene A n : C [ n lows: If k n,thenwedenea n j[ n + n 7 + 1n [ := A n j[ kn + n kn + n + 1n [ : j[ n + n 7 + 1n [ := (1) [5 n 5 n ], i.e. Otherwise we dene A n the output equals the piece ofscenery which is constantly equal to 1. In the remainder of the article we will prove that A n specied by Theorem 1. 5 Some Lemmas For k N we dene the events E n k := ( k1 i= E n k := S in Note that E n k = f k kg. Z i = ) S kn > Ln for all i<k : fullls its task Lemma 3 For n N,letF(n) be the-algebra generated by the whole scenery and the random walk up to time n: F(n) :=( S(i) i [ n]). There existsa constant c 3 > such that for all n N and all k n h E n k i c F n (k 1) + n c3 n n : roof. By the local central limit theorem (see e.g. [], Theorem (5.), page 13), the probability that the random walk after n n steps hits a set containing at most 4Ln +1 n points is bounded by c 4 n 3 4Ln n = c3 n n here c 4 > is a constant independent ofn and c 3 =4Lc 4. The claim follows. The next lemma gives information about the tail probability of k. We need some notation: For k N,let F (k) denote the tail probability: F (k) := k>k : () 5

7 We set " (n ) := s max 1 " 1 (n ) (7) n k n := min k : F (k) " (n ) : (8) Lemma 4 There exists c 5 such that k n n for all n c 5. roof. For k N,we dene the Bernoulli random variable Zk which is equal to 1 if and only if Z k =1orE n k does not hold. We have that ( n k= Z k > ) n k= E n k k n thus n F (n )= k>n k= h i c E n k + ( n k= Z k = )! : (9) By Lemma 3, n k= h i c E n k (n +1)c 3 n n : (1). The sequence ( Zk k ) is adapted to the ltra- We dene the -algebra G k := Z i S(in ) S(jn ) (S(in )+z) i<j k +1 z Ln Ln tion (G k k ). Furthermore, by the denition of Zk, ( Zk =1jG k1 ) = 1E n k (Z k =1jG k1 )+1 1E n k E B (n ) h i c E n k here 1A denotes the indicator function of the set A. For the rst equality we used E n k G k1. For the last inequality we used that Z k is measurable with respect to (S(kn )+z) z Ln Ln which is independent of G k1. We abbreviate p n := E B (n ). The preceding estimate yields n k= Z k =! = E = " 1 ( n k= "( n1 k= (1 p n ) Z k = Z k = ( n1 k= ) ) n k= Z k = n k= ) E n k E n k n 1 k= #! Zn = # Gn1! E n k

8 here E denotes the expectation with respect to. Using an induction argument, we conclude n k= Z k =! (1 p n ) n : (11) In order to obtain a lower bound for p n, rst note (z) =18z n 11 n11 =(1=C) n Furthermore, by Doob's inequality (see e.g. [], page 5), max js(i)j >n 11 i[ n ] n Var(S(n )) = c n with some constant c >. Thus, we obtain for all n suciently large p n = E B (n ) (z) =18z n 11 n11 max js(i)jn 11 i[ n ] C n11 1 (1 c n ) Cn11 : (1) It follows from (11) and (1) that n k= Z k =! 1 C n11 n exp n C n11 : Using the denition of n (1), we see that the right-hand side is bounded above by 1 n 1= for all n suciently large. Combining this with (9) and (1) yields for all n suciently large F (n ) (n +1)c 3 n n + 1 n 1= n 1= " (n ) for the last inequality we used the denition of " (n ) (7). The claim follows from the denition of k n. We dene E n := S in 4 S kn 4 > Ln E n := k k n E n : Lemma 5 The following holds: roof. We observe that lim n!1 ([En ]c )=: for all i< k ([E n ]c ) = k>kn + k k n E n c : 7

9 By the denition of F and kn, k>kn = F (kn ) " (n ), which convergestoasn!1 recall the denition of " (n )and(4). Using Lemmas 3 and 4, we obtain for all n suciently large k k n k E n c n i= h i c E n i c3 n n (n +1) which convergestoasn!1by the denition of n (1). The claim follows. roof of Theorem 1 roof of Theorem 1. The following holds: ([E n ini works ]c ) ([E n ini works ]c E n )+ ([En ]c ) : By Lemma 5, the second term on the right-hand side converges to as n!1. It suces to prove that the same is true for the rst term. We observe that ([E n ini works ]c E n )= kn k= By the denition of the shift, we have form [E n ini works ]c E n j k = k k = k m ( S) =( ( + S (m)) S( + m) S (m)) : Consequently, we obtain for n suciently large and k k n kn n (E n recon) k = k E n ini works k = k (13) here we used that if k = k, then the rst observation which A n uses is kn + n and for all n suciently large, L n1 S kn + n kn + n L k n n + n L n 7 + n because the random walker starts at the origin and jumps in each step at most a distance of L. Hence the reconstruction starts not at the origin, but at position S kn + n which has a distance n 1 from the origin. This is why dierent pieces of scenery of are concerned in the denitions of Erecon n and. Thus (13) yields E n ini works ([E n ini works ]c E n ) (14) k n k= k n k= h kn n (E n recon) h kn n (E n recon) 8 i c n E k = k i c f k = kge n k = k k = k : (15)

10 Note that E n k = f k kg and E n k = E n fk = kg. Consequently, for k k n, f k = kge n =kn (E B (n )) E n k E n k : (1) By the Markov property, we know that S t + kn S kn t is independent of S(t) t [ kn ]. The event kn (EB (n )) depends only on S t + kn S kn t and S(kn )+z z Ln Ln because the random walker can jump in each step at most a distance of L. On the other hand, wehavethate n k E n k only depends on S(t) t kn and the scenery S(kn )+z z Ln. Since the scenery is Ln i.i.d., the event kn (EB (n )) is independent ofe n k E n k. For any events A B C with the property thata and B are independent and (B) > the following inequality holds: (CjA B) (CjA) (B) : In our case this yields together with (1): = = i h kn n c (E recon) n fk = kge n i h kn n c (E recon) n kn (E B (n )) E n k h c kn n (E n recon)i kn (EB (n )) h n (E n E n k E n k c recon)i EB (n ) E n k E n k = E n k (17) E n k E n k " 1 (n ) (18) for the second but last inequality we used that the shift preserves the measure by Lemma 4.1 of [1] for the last equality we used denition (). Using the monotonicity of F and the denition of kn,we obtain for all k k n E n k = k k = F (k 1) F (kn 1) " (n ) : Combinig the last inequality with Lemma 3 and the fact E n k F(t) for t = n (k 1) + n,we obtain h h E n k ii E n k F(t) i 1 c 3 n n he n k 1 c 3 n n " (n ) : For n big enough we get that 1 c 3 n n > 1=. In that case we conclude E n k E n k " (n ) =: 9

11 Combining the last estimate with (18), we obtain h kn (E n recon) i c f k = kge n " 1 (n ) " (n ) : By the denition of " (n )(7),wehave " (n ) p " 1 (n ). Thus, Using (15) we get h kn (E n recon) i c f k = kge n ([E n ini works ]c E n ) p " 1 (n ): p " 1 (n ): It follows from (4) that lim n!1 ([E n ini works ]c E n )=. This completes the proof of Theorem 1. References [1] I. Benjamini and H. Kesten. Distinguishing sceneries by observing the scenery along a random walk path. J. Anal. Math., 9:97{135, 199. [] R. Durrett. robability: Theory and Examples. Duxbury ress, Second edition, 199. [3] C. D. Howard. Orthogonality of measures induced by random walks with scenery. Combin. robab. Comput., 5(3):47{5, 199. [4] C. D. Howard. Distinguishing certain random sceneries on Z via random walks. Statist. robab. Lett., 34():13{13, [5] H. Kesten. Detecting a single defect in a scenery by observing the scenery along a random walk path. In It^o's stochastic calculus and probability theory, pages 171{183. Springer, Tyo, 199. [] H. Kesten. Distinguishing and reconstructing sceneries from observations along random walk paths. In Microsurveys in discrete probability (rinceton, NJ, 1997), pages 75{83. Amer. Math. Soc., rovidence, RI, [7] E. Lindenstrauss. Indistinguishable sceneries. Random Structures Algorithms, 14(1):71{8, [8] M. Lowe and H. Matzinger. Reconstruction of sceneries with correlated colors. Eurandom Report 99-3, [9] M. Lowe and H. Matzinger. Scenery reconstruction in two dimensions with many colors. Eurandom Report 99-18, [1] M. Lowe, H. Matzinger, and F. Merkl. Reconstructing a multicolor random scenery seen along a random walk path with bounded jumps. Eurandom Report 1-3, 1. 1

12 [11] H. Matzinger. Reconstructing a -color scenery by observing it along a simple random walk path with holding. hd thesis, Cornell University, [1] H. Matzinger. Reconstructing a three-color scenery by observing it along a simple random walk path. Random Structures Algorithms, 15():19{7, [13] H. Matzinger. Reconstructing a -color scenery by observing it along a simple random walk path. Eurandom Report -3,. [14] H. Matzinger and S. W. W. Rolles. Reconstructing a random scenery observed with random errors along a random walk path. reprint. 11