A local limit theorem for a transient chaotic walk
in a frozen environment
Abstract
This paper studies particle propagation in a onedimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle’s initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a onedimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk’s probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over a coarser diffusive space scale is Gaussian.
Keywords: random walk in random environment, quenched random walk, random media, local limit theorem, extended dynamical system
AMS 2000 Subject Classification: 60K37; 60F15; 37H99; 82C41; 82D30
1 Introduction
1.1 A chaotic dynamical system
This paper studies a particle moving in a continuous inhomogeneous medium which is composed of a linear chain of cells modeled by the unit intervals of the positive real line. Each interval is assigned a label and a map which determines the dynamics of the particle as long as the particle remains in the interval. The sequence of labels , called an environment, is assumed to be either nonrandom, or a realization of a random sequence that is frozen during the particle’s lifetime.
We are interested in the case in which the local dynamical rules are chaotic in the sense that the distance between two initially nearby particles grows at an exponential rate. More concretely, we shall focus on a model where a particle located at at time jumps to . Here and is the piecewise affine map from onto such that and . The dynamical system generated by the local rules is compactly expressed by , where the global map on the positive real line is defined by
(1.1) 
and denotes the integral part of , see Figure 1.
The above model belongs to the realm of extended dynamical systems, a somewhat vaguely defined yet highly active field of research (e.g. Chazottes and Fernandez [9]). Telltale characteristics of such systems are a noncompact or highdimensional phase space and the lack of relevant finite invariant measures. Our principal motivation is to study the impact of environment inhomogeneities on the longterm behavior of extended dynamical systems. Concrete models include neural oscillator networks (Lin, SheaBrown, and Young [27]) and the Lorentz gas with randomly placed scatterers (Chernov and Dolgopyat [11]; Cristadoro, Lenci, and Seri [12]), to name a few. In this paper, we shall restrict the analysis to the affine dynamical model in (1.1), to keep the presentation simple and clear.
1.2 Random initial data
Because the local maps are chaotic, predicting the particle’s future location with any useful accuracy over any reasonably long time horizon would require a precise knowledge of its initial position — a sheer impossibility in practice. Therefore, it is natural to take the statistical point of view and study the stochastic process defined by
(1.2)  
To analyze the time evolution of the above process, we must impose some regularity conditions on the environment. In particular, those conditions guarantee ballistic motion, and one might guess that the distribution approaches Gaussian in the long run. To test this hypothesis, we have plotted in Figure 2 numerically computed histograms of at time in two frozen environments, using the intervals as bins. Rather surprisingly, the histograms do not appear Gaussian. A similar phenomenon was recently observed by Simula and Stenlund [33, 34].
1.3 Summary of main results
The main contribution of the paper is to explain the emergence of the histograms in Figure 2. This is accomplished by first reducing the continuum dynamical system to a unidirectional random walk on the integers (Theorem 1), and then deriving a local limit law (Theorem 4) that completely explains the behavior observed in Figure 2. As a byproduct, we also obtain a law of large numbers (Theorem 3) and a central limit law (Theorem 5) for the walk. These limit laws are valid for all frozen environments — random or nonrandom — which satisfy certain statistical regularity properties. We also devote a separate section to the analysis of random environments, where we show (Theorem 7) that the three aforementioned limit laws are valid for almost all realizations of a stationary random environment under suitable moment and mixing conditions. Because the random walk is unidirectional (it never steps backwards), limit theorems for its hitting times are immediate consequences of classical limit laws for independent random variables. Translating the limit theorems of the hitting times into limit theorems of the walk location form the main task in proving the results; see Section 3.3 for a general outline of the proofs.
1.4 Related work
We shall discuss here only literature most closely related to transient onedimensional random walks in quenched random environments; for a broad picture of the theory of random walks in random environments, see e.g. Bolthausen and Sznitman [6], Sznitman [38], and Zeitouni [40]. Laws of large numbers and averaged central limit theorems for random walks in random environments have been known already for a long time (e.g. Solomon [35]; Kesten, Kozlov, and Spitzer [25]), whereas the literature on quenched central limit theorems is more recent. Buffet and Hannigan [8] proved a quenched central limit theorem for a pure birth process in an independently scattered random environment under moment conditions later relaxed by Horváth and Shao [21, 22]; this model is a direct continuoustime analogue of the random walk studied here. Quenched central limit theorems for more general onedimensional transient random walks were proved only very recently, independently by Goldsheid [17] and Peterson [30] (see also Alili [1] for a result concerning a special quasiperiodic environment). RassoulAgha and Seppäläinen [32] obtained a similar result for multidimensional random walks with a forbidden direction, which in the onedimensional case corresponds to the unidirectional walk analyzed in this paper. Dolgopyat, Keller, and Liverani [14] have obtained a quenched central limit theorem for environments changing in time and space.
Our approach differs from most earlier works in that we separately analyze the two degrees of randomness involved in random walks in quenched random environments. In the first part, we extract a set of statistical regularity properties for a given environment that are sufficient for proving the limit laws, while treating the environment as nonrandom. In the second part, we derive conditions for the probability distribution of the random environment that yield almost surely regular realizations. A key result for the second part is a law of large numbers for the moving averages of a stationary sequence (Lemma 2), which is proved with the help of Peligrad’s extension [29] of the Baum–Katz theorem [3] (see Bingham [4] for a nice survey). The major advantage of this approach is a clarified picture on how different sources of randomness affect the random walk’s behavior in quenched random environments.
Local limit theorems for random walks in homogeneous environments (e.g. Spitzer [36]) can be proved as simple consequences of Gnedenko’s classical theorem (e.g. [16, Chapter 9] or [15, Section 3.5]), because such walks are just sums of independent random variables. A generalization for a periodic environment was derived by Takenami [39], and for a periodic graph recently by Kazami and Uchiyama [24]. In contrast, local limit theorems for random walks in aperiodic inhomogeneous environments appear nonexistent. To the best of our knowledge, Theorem 4 and Theorem 7 are the first local limit laws concerning transient random walks in aperiodic nonrandom or quenched random environments. Although our analysis is restricted to a very special instance of a random walk, the model is still rich enough to capture several interesting phenomena, such as the need for nonlinear centering and a nonGaussian modulating factor, and we believe that the results could serve as useful benchmarks when testing hypotheses concerning more general random walks.
Regarding extended dynamical systems, we have found two earlier local limit theorems, both corresponding to homogeneous environments. Szász and Varjú [37] have considered Lorentz processes with periodic configurations of scatterers, while Bardet, Gouëzel, and Keller [2] study rather different type of systems: small (possibly inhomogeneous) perturbations of weakly coupled, translation invariant, coupled map lattices; see Nagaev [28] and Guivarc’h [20] for some of the original techniques. Let us finally stress that for more classical, probability measure preserving, dynamical systems, various types of limit theorems have been proved for many decades. Yet such systems, too, continue to be studied vigorously, with important recent developments (e.g. Chazottes and Gouëzel [10]; Gouëzel [18, 19]).
1.5 Organization of the paper
The rest of the paper is organized as follows. Section 2 presents the main results. The proofs for nonrandom environments are given in Section 3, and the proofs for quenched random environments in Section 4. Section 5 concludes the paper, and Appendix A contains basic facts on generalized inverses of increasing sequences.
2 Main results
2.1 Representation as a random walk
The distribution of the dynamical system (1.2) at any time instant can be completely characterized in terms of the following simple unidirectional random walk (discretetime pure birth process) on the integers. Given an environment , let be a random walk in having the initial state and transitions
(2.1) 
We denote by the distribution of the walk in the path space . Note that if the environment is a realization of a random sequence, the process can be identified as a random walk in a random environment, and the distribution is usually called the quenched law of the random walk. The expectation and variance with respect to are denoted by and , respectively. When presenting general facts in probability theory, we write and for the measure and expectation.
Theorem 1.
Proof.
The proof follows by induction and Lemma 2 below. ∎
Lemma 2.
Let be a positive random variable such that (i) and are independent, and (ii) is uniformly distributed in . Then the same is true for , and moreover,
(2.2) 
whenever .
Proof.
Denote , where is a positive random integer independent of , and is uniformly distributed in . Assume first that for some nonrandom integer . Then . Moreover, a simple calculation based on the definition of shows that for all ,
(2.3) 
Hence satisfies (i), (ii), and (2.2) in the case where is nonrandom. The general case follows by conditioning on . ∎
2.2 Limit theorems for regular frozen environments
In this section we shall analyze the random walk in a fixed, sufficiently regular environment, which may either be nonrandom, or a realization of a random sequence. More precisely, we shall in general assume that the environment is such that
(2.4) 
for some ; and
(2.5)  
(2.6) 
for some constants and . Moreover, we assume that
(2.7) 
and that the environmental moving averages satisfy
(2.8) 
for all , where . (In the special case with it suffices to use ). Concrete examples of environments that satisfy the above regularity conditions shall be given in Section 2.3.
The quantity represents the mean sojourn time of the particle in the interval (see Section 3.1 for more details). Therefore, the constant appearing in (2.5) may be interpreted as the inverse of the particle’s traveling speed. The following result confirms this intuition.
The main result of the paper is the following limit theorem for the random walk defined by (2.1), or alternatively (by virtue of Theorem 1), for the dynamical system defined by (1.2).
Theorem 4 (Local limit theorem).
Two features in Theorem 4, which distinguish it from classical limit theorems, call for special attention. First, the centering factors depend on the environment, and are in general nonlinear functions of . Second, the modulating factor in (2.10) causes the asymptotic shape of the probability mass function of to be nonGaussian. This modulating factor explains the behavior observed in Figure 2.
In contrast, when looking at the probability distribution of the walk over a coarser diffusive space scale, the nonGaussian modulating factor in Theorem 4 averages out asymptotically, and we end up with a standard Gaussian limiting distribution.
Theorem 5 (Central limit theorem).
Remark 6.
The centering factor may be identified as a generalized inverse (Appendix A) of the function , where denotes the hitting time of the walk into site (Section 3.1). A quick inspection of the proofs in Section 3 shows that Theorem 4 remains true if is replaced by , where is as in (2.4); use the inequality for and the proof of Lemma 6 for . Moreover, Theorem 5 remains true if is replaced by .
2.3 Limit theorems for quenched random environments
In this section we assume that the environment is a realization of a stationary random sequence in , and denote by its distribution on . The expectation with respect to is denoted by . We shall assume that
(2.13) 
To guarantee that the environmental averages converge to their mean values rapidly enough, we assume that
(2.14) 
where the mixing coefficients are defined by
(2.15) 
and where and (e.g. Bradley [7]).
The following result summarizes three limit theorems for the quenched random walk in a stationary strongly mixing random environment.
Theorem 7.
Especially, the limit laws summarized by Theorem 7 hold in the following cases:

Independently scattered stationary environments (environments where the site labels are independent and identically distributed).

Uniformly ergodic environments as discussed in Goldsheid [17].

Environments which are realizations of finitestate irreducible aperiodic stationary Markov chains, or more general Markov chains satisfying Doeblin’s condition (e.g. [7]).
Although in many applications it is natural to assume that the environment is stationary, the limits of Theorem 7 remain valid under looser conditions, as is clear from the results of Section 2.2.
Alternative versions of the law of large numbers and the central limit law in Theorem 7, where the centering factors are replaced by , can be proved as consequences of Theorems 3.1 and 5.4 in RassoulAgha and Seppäläinen [32], if we additionally assume that
This socalled nonnestling assumption is close in spirit to the uniform ellipticity of nearestneighbor random walks in random environments; in the context of our model it corresponds to the special case in (2.4). Although we believe that the local and central limit laws in Theorems 4, 5, and 7 remain generally valid also for the alternative centering , we prefer to use the centering defined in (2.11), because these factors are easily computed from the environment. Analogous central limit laws for nearestneighbor walks were recently independently found by Goldsheid [17] and Peterson [30].
If we were only interested in the law of large numbers (2.9), we could do with less assumptions in Theorem 7. For example, as our proof in Section 4 shows, the moment condition (2.13) would only be needed for . The mixing assumption (2.14) could be relaxed as well, see for example Bingham [4] for more details.
3 Proofs for regular nonrandom environments
This section is devoted to proving Theorems 3–5 for the random walk in an environment that satisfies the regularity assumptions (2.4) – (2.8). The environment shall be fixed once and for all during the whole section — here we do not care whether it is a realization of a random sequence or not.
The section is organized as follows. Section 3.1 describes some preliminaries on the hitting times of the walk, and Section 3.2 gives the proof of the law of large numbers. The proof of the local limit theorem is split into Sections 3.3–3.7, and the proof of the central limit theorem is in Section 3.8.
3.1 Hitting times of the walk
We denote the hitting time of into site by , and the sojourn time at site by . Because the walk never moves backwards, the equivalence
(3.1) 
is valid for all and . Moreover, the sojourn times are independent, and has a geometric distribution on with success probability . Hence the mean and the variance of are given by and , respectively. The mean and the variance of are denoted by and , so that
(3.2) 
The following result transforms the realizationbyrealization relationship (3.1) into one concerning the probability mass functions.
Lemma 1.
For any environment and any ,
Proof.
Note that for all . Because and are independent, we find by applying (3.1) and conditioning on that
This implies that claim, because and almost surely. ∎
3.2 Proof of Theorem 3
By (2.4) we see that for some constants and , so that , which implies as a consequence of Kolmogorov’s variance criterion (e.g. [23, Cor. 4.22]) that almost surely. By (2.5), we conclude that the hitting times of the walk satisfy the following law of large numbers:
(3.3) 
Further, by (3.1), we see that , where denotes the generalized inverse of the sequence defined in Appendix A. Lemma 1 together with (3.3) now shows that almost surely, and the proof of Theorem 3 is complete. ∎
3.3 Outline of proof for the local limit theorem
The starting point of the proof is Lemma 1 in Section 3.1, which reduces the problem into analyzing the probability mass function of the hitting times . Because is a sum of independent random variables, we may apply a classical local limit theorem (e.g. Petrov [31]) in Section 3.4 to conclude that for large values of , where
(3.4) 
and and denote the mean and the variance of given by (3.2).
In the rest of the proof we need to transform the Gaussian density of the time variable into a Gaussian density of the space variable. This will be accomplished in two steps. We show in Section 3.5 that , where
(3.5) 
and further in Section 3.6 that , where
(3.6) 
is the Gaussian density appearing in the statement of Theorem 4 (we write in place of for convenience).
The approximation is a subtle part in the argument, where we have been guided by the following intuition. By approximating , we obtain , where . Therefore, the difference of the square roots of the exponents in and can be approximated by
Condition (2.8) has been tailored to guarantee that the above difference is small for values of such that .
An essential technical complication is the fact that the modulating factor in (2.10) (see also Lemma 1) is unbounded. To overcome this difficulty, we have strived to obtain sharp estimates. Identifying the set of reasonable sufficient assumptions (2.4) – (2.8) has played a crucial role in the proof.
3.4 Local limit theorem for the hitting times
By applying a classical local limit theorem for sums of independent random variables (Petrov [31, Theorem VII.5]; see also Davis and McDonald [13]), we obtain the following local limit theorem for the hitting times.
Lemma 2.
Proof.
Observe first that by (2.6),
and that
because of and (2.7). Therefore, to apply [31, Theorem VII.5], it suffices to verify (note that for all and ) that
(3.8) 
By writing , where and , and applying Hölder’s inequality with conjugate exponents and , we see that
After dividing both sides above by , and applying (2.6) and (2.7), we see that
which implies (3.8), because . The claim now follows by applying [31, Theorem VII.5] and recalling that by (2.6). ∎
The following result is an analogue of Lemma 2, where the time variable instead of the space variable tends to infinity.
Lemma 3.
Proof.
Fix an , and note that
(3.9) 
by Lemma 2. Therefore, we only need to analyze and for . As a preliminary, note that, because and by (2.5), we may fix a constant and an integer such that for all .
Assume now that and . Then , and moreover, , where is finite by (2.6). Therefore,
(3.10) 
where . A rough estimate together with Chebyshev’s inequality shows that
so by (3.10), we conclude that
(3.11) 
By applying (3.10) once more, we see that
(3.12) 
where is strictly positive by (2.6). The proof is now completed by combining the estimates (3.11) and (3.12) with (3.9). ∎
3.5 Variance of the hitting times
Lemma 4.
Proof.
The proof is split into two parts according to whether or not , where is a large constant to be determined later.
(i) Assume that is such that . Using the triangle inequality and the inequality , we see that
Consequently, assuming that is large enough so that ,
where .
Observe next that, assuming is large enough so that ,
(3.14) 
where . Further,
Now using (3.14) we find that for . Therefore,
where
As a consequence,
(3.15) 
for all large enough , where . By combining (2.5) and (2.6), we find that . Because by (3.14), we conclude that the right side above tends to zero as .
(ii) Assume now that is such that . Note that , where and are finite and strictly positive by (2.6). Therefore, the exponent in the definition of is bounded by
Consequently,
(3.16) 
where and . For the function we immediately see that
(3.17) 
where and . The proof is now finished by choosing large enough so that and , and combining the estimates (3.16) and (3.17) with (3.15). ∎
3.6 From hitting times to walk locations
Lemma 5.
Proof.
We will show the claim by treating separately the cases and , where is as in (2.8), and is a large constant to be determined later.
(i) Assume that is such that . Using the inequality , we see that
where , and . Note that
(3.18) 
where the latter inequality is due to . To analyze the right side of (3.18), observe that, because , we see using (2.8) that
The above limiting relation also shows (substitute ) that . Because (by (2.5) and Lemma 1), it follows that
and therefore,
(3.19) 