A FUNCTIONAL LIMIT THEOREM FOR A 2D-RANDOM WALK WITH DEPENDENT MARGINALS

We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has dif-ferent horizontal and vertical ﬂuctuations leading to diﬀerent normalizations in the functional limit theorem, with a non-Gaussian horizontal behavior. We also prove that the horizontal and vertical components are not asymptotically independent.


Introduction
The study of random walks on oriented lattices has been recently intensified with some physical motivations, e.g. in quantum information theory where the action of a noisy channel on a quantum state is related to random walks evolving on directed graphs (see [2,3]), but they also have their own mathematical interest. A particular model where the simple random walk becomes transient on an oriented version of Z 2 has been introduced in [3] and extended in [4] where we have proved a functional limit theorem. In this model, the simple random walk is considered on an orientation of Z 2 where the horizontal edges are unidirectional in some i.i.d. centered random way. This extra randomness yields larger horizontal fluctuations transforming the usual normalization in n 1/2 into a normalization in n 3/4 , leading to a non-Gaussian horizontal asymptotic component. The undirected vertical moves still have standard fluctuations in n 1/2 that are thus killed by the larger normalization in the result proved in [4] (Theorem 4), yielding a null vertical component in the limit. If these horizontal and vertical asymptotic components were independent, one could state this functional limit theorem with an horizontal normalization in n 3/4 and a vertical one in n 1/2 , but it might not be the case.
Here, we prove this result without using independence and as a complementary result we indeed prove that these two asymptotic components are not independent.

Model and results
The considered lattices are oriented versions of Z 2 : the vertical lines are not oriented but the horizontal ones are unidirectional, the orientation at a level y ∈ Z being given by a Rademacher random variable ǫ y = ±1 (say left if the value is +1 and right if it is −1). We consider here the i.i.d. case where the random field ǫ = (ǫ y ) y∈Z has a product law P ǫ = ⊗ y∈Z P y ǫ defined on some probability space (A, A, Q) with marginals given by P y ǫ [±1] = Q[ǫ y = ±1] = 1 2 .
Definition 1 (Oriented lattices) Let ǫ = (ǫ y ) y∈Z be a sequence of random variables defined as previously. The oriented lattice L ǫ = (V, A ǫ ) is the (random) directed graph with (deterministic) vertex set V = Z 2 and (random) edge set A ǫ defined by the condition that for These orientations will act as generalized random sceneries and and we denote by W = (W t ) t≥0 the Brownian motion associated to it, i.e. such that (2.1) In this paper, the notation D =⇒ stands for weak convergence in the space D = D([0, ∞[, R n ), for either n = 1, 2, of processes with càdlàg trajectories equipped with the Skorohod topology. 1 For every realization of ǫ, one usually means by simple random walk on L ǫ the Z 2 -valued Markov chain M = M (1) n , M (2) n defined on a probability space (Ω, B, P), whose (ǫ-dependent) transition probabilities are defined for all (u, v) ∈ V × V by In this paper however, our results are also valid when the probability of an horizontal move in the direction of the orientation is 1 − p ∈ [0, 1] instead of 1 3 , with probabilities of moving up or down equal thus to p 2 . We write then m = 1−p p for the mean of any geometric random variable of parameter p, whose value is m = 1 2 in the standard case p = 2 3 . We also use a self-similar process ∆ = (∆ t ) t≥0 introduced in [5] as the asymptotic limit of a random walk in a random scenery, formally defined for t ≥ 0 by where L = (L t ) t≥0 is the local time of a standard Brownian motion B = (B t ) t≥0 , related to the vertical component of the walk and independent of W . We also denote for all t ≥ 0 The following functional limit theorem has been proved in [4]: We complete here this result with the following theorem: and the asymptotic components ∆

Random walk in generalized random sceneries
We suppose that there exists some probability space (Ω, F, P) on which are defined all the random variables, like e.g. the orientations ǫ and the Markov chain M .

Embedding of the simple random walk
We use the orientations to embed the 2d-random walk on L ǫ into two different components: a vertical simple random walk and an horizontal more sophisticated process.

Vertical embedding: simple random walk
The vertical embedding is a one dimensional simple random walk Y , that weakly converges in D to a standard Brownian motion B: The local time of the walk Y is the discrete-time process N (y) = (N n (y)) n∈N canonically defined for all y ∈ Z and n ∈ N by That is in particular characterized by the following behavior established in [5]: For any reals a < b, the fraction of time spent by the process One is then particularly interested in analogous quantities for the Brownian motion (B t ) t≥0 , i.e. in a local time L t (x) and in a fraction of time spent in [a, b) before t. If one defines naturally the former fraction of time to be Λ t (a, b) = t 0 1 [a≤Bs<b] ds then ( [6]) one can define for all x ∈ R such a process L t (x) t>0 , jointly continuous in t and x, and s.t., To prove convergence of the finite-dimensional distributions in Theorem 2, we need a more precise relationship between these quantities and consider the joint distribution of the fraction of time and the random walk itself, whose marginals are not necessarily independent.
Proof: For t ≥ 0, define the projection π t from D to R as π t (x) = x t . From [5], the map is continuous on D([0, T ]) in the Skorohod topology for any T ≥ t for almost any sample point of the process (B t ) t≥0 . Moreover, since almost all paths of the Brownian motion (B t ) t≥0 are continuous at t, the map x → π t (x) is continuous at a.e. sample points of the process (B t ) t≥0 . So, for any t ≥ 0, for any a, b ∈ R and any θ 1 ∈ R, θ 2 ∈ R, the map to the process (B t ) t≥0 implies then the convergence of the law of This proves the lemma using the characteristic function criterion for convergence in distribution. ⋄

Horizontal embedding: generalized random walk in a random scenery
The horizontal embedding is a random walk with geometric jumps: consider a doubly infinite family (ξ (y) i ) i∈N * ,y∈Z of independent geometric random variables of mean m = 1−p p and define the embedded horizontal random walk X = (X n ) n∈N by X 0 = 0 and for n ≥ 1, with the convention that the last sum is zero when N n−1 (y) = 0. Define now for n ∈ N the random time T n to be the instant just after the n th vertical move, Precisely at this time, the simple random walk on L ǫ coincides with its embedding. The following lemma has been proved in [3] and [4]:

Random walk in a random scenery
We call X a generalized random walk in a random scenery because it is a geometric distorsion of the following random walk in a random scenery Z = (Z n ) n∈N introduced in [5] with From the second expression in terms of the local time of the simple random walk Y , it is straightforward to see that its variance is of order n 3/2 , justifying the normalization in n 3/4 in the functional limit theorem established in [5]. There, the limiting process ∆ = (∆ t ) t≥0 of the sequence of stochastic processes n − 3 4 Z [nt] t≥0 is the process obtained from the random walk in a random scenery when Z is changed into R, the random walk Y into a Brownian motion B = (B t ) t≥0 and the random scenery (ǫ y ) y∈Z into a white noise, time derivative in the distributional sense of a Brownian motion (W (x)) x∈R . Formally replacing N n (x) by L t (x), the process ∆ can be represented by the stochastic integral Since the random scenery is defined on the whole Z axis, the Brownian motion (W (x)) x∈R is to be defined with real time. Therefore, one introduces a pair of independent Brownian motions (W + , W − ) so that the limiting process can be rewritten In addition to its existence, Kesten and Spitzer have also proved the We complete this result and consider the (non-independent) coupling between the simple vertical random walk and the random walk in a random scenery and prove:

Strategy
The main strategy is to relate the simple random walk on the oriented lattice L ǫ to the random walk in random scenery Z using the embedded process (X, Y ). We first prove the functional limit Theorem 4 by carefully carrying the strategy of [5], used to prove Theorem 3, for a possibly non independent couple (Z, Y ). This result extends to the embedded process (X, Y ) due to an asymptotic equivalence in probability of X with a multiple of Z. Theorem 2 is then deduced from it using nice convergence properties of the random times (3.7) and self-similarity. Eventually, we prove that the asymptotic horizontal components of these twodimensional processes are not independent, using stochastic calculus techniques.

Proof of Theorem 4
We focus on the convergence of finite dimensional distributions, because we do not really need the tightness to prove our main result Theorem 2. It could nevertheless be proved in the similar way as the tightness in Lemma 7, see next section.
Proof: We first identify the finite dimensional distributions of ∆ t , B t t≥0 .
Lemma 4 For any distinct t 1 , . . . , t k ≥ 0 and θ 1 , . . . , θ k ∈ R 2 , the characteristic function of the corresponding linear combination of ∆ t , B t is given by j L t j (x) being continuous, almost surely with compact support, for almost all fixed sample of the random process (B t ) t , the stochastic integrals are independent Gaussian random variables, centered, with variance +∞ 0 k j=1 θ (1) Therefore, for almost all fixed sample of the random process B, k j=1 θ (1) j ∆ t j is a centered Gaussian random variable with variance given by Then we get Hence we have expressed the characteristic function of the linear combination of (∆ t , B t ) t≥0 in terms of B and its local time only. We focus now on the limit of the couple 1 n 3/4 Z [nt] , 1 n 1/2 Y [nt] t≥0 when n goes to infinity and introduce for distinct t j ≥ 0 and θ j ∈ R 2 the characteristic function By independence of the random walk Y with the random scenery ǫ, one gets where λ(θ) = E e iθǫy is the characteristic function of the orientation ǫ y , defined for all y ∈ Z and for all θ ∈ R. Define now for any θ j ∈ R 2 and n ≥ 1, Lemma 5 lim n→∞ φ n (θ 1 , . . . , θ k ) − ψ n (θ 1 , . . . , θ k ) = 0.
The first term tends to zero in virtue of Lemma 1. The second term also vanishes in the limit because one has λ(θ) ∼ 1 − θ 2 2 as |θ| → 0. Thus Lemma 5 is proved. ⋄ The asymptotic behavior of φ n will be this of ψ n and we identify now its limit with the characteristic function of the linear combination of ∆ t , B t t≥0 in the following: Lemma 6 For any distinct t 1 , . . . , t k ≥ 0 and θ 1 , . . . , θ k ∈ R 2 , the distribution of Proof: We proceed like in [5] where a similar result is proved for the horizontal component; although the convergence holds for each component, their possible non-independence prevents to get the convergence for the couple directly and we have to proceed carefully using similar steps and Lemma 2. We decompose the set of all possible indices into small slices where sharp estimates can be made, and proceed on them of two different limits on their sizes afterwards. Define, in a slice of size a(l, n) = τ l √ n, l ∈ Z, an average occupation time by Consider δ(l, n) = a(l + 1, n) − a(l, n) and write First step: We first show that A(τ, M, n) tends in probability to zero as n → ∞, for a fixed τ in the slice of length δ(l, n). Fix also M and n and write and similarly, Thus, using Lemma 1 and 3 from [5], we have for large n, We will afterwards consider the limit M τ 3/2 goes to zero to approximate the stochastic integral of the local time L t , and this term will then go to zero. Moreover, we have From item b) of Lemma 1 in [5] , we can choose M τ so large that P U (τ, M, n) = 0 is small. Then, we have proved that for each η > 0, we can choose τ, M and large n such that Second step: From Lemma 2, V (τ, M, n) converges in distribution, when n → ∞, to The function x → L t (x) being continuous and having a.s compact support, Putting together Lemma 4, 5 and 6 gives Proposition 1, that proves Theorem 4. ⋄

Proof of Theorem 2
We get the convergence of Theorem 2 from Theorem 4 and Lemma 3 and focus first on the embedded process (X, Y ): Proof: We first prove the tightness of the family. The second component is tight in D (see Donsker's theorem in [1]), so to prove the proposition we only have to prove the tightness of the first one in D. By Theorem 13.5 of Billingsley [1], it suffices to prove that there exists K > 0 such that for all t, t 1 , (4.9) Using Cauchy-Schwarz inequality, it is enough to prove that there exists K > 0 such that for all t 1 ≤ t, for all n ≥ 1, Since the ǫ ′ s are independent and centered, we have From item d) of Lemma 1 in [5], as n tends to infinity, and there exists some constant K > 0 such that We get the tightness of the first component by dividing X n by n 3/4 , and eventually the tightness of the properly normalized embedded process.
To deal with finite dimensional distributions, we rewrite X n = X n + mZ n−1 with Using the L 2 −convergence proved in the proof of Proposition 2 in [4], . One concludes then using Theorem 4. ⋄ In the second step of the proof of Theorem 2, we use Lemma 3 of [4] and that M Tn = M (1) Tn = (X n , Y n ) for any n with T n n n→∞ −→ 1 + m , P − almost surely and the self-similarity of the limit process ∆ (index 3/4) and of the Brownian motion B (index 1/2). Using the strict increasing of (T n ) n∈N , there exists a sequence of integers (U n ) n which tends to infinity and such that T Un ≤ n < T Un+1 . More formally, for any n ≥ 0, U n = sup{k ≥ 0; T k ≤ n}, (U [nt] /n) n≥1 converges a.s. to the continuous function φ(t) := t/(1 + m), so from Theorem 14.4 from [1], Using Lemma 7, the processes (m∆ φ(t) , B φ(t) ) t and [nt] = M and M (1) , so By remarking that for every T > 0, (1) is enough to prove that ∆ 1 and B 1 are not independent and we use that conditionnally to (B t ) 0≤t≤1 , the random variable ∆ 1 is the sum of the stochastic integrals Proof: For every x ∈ R, define J ε (x) = 1 2ε 1 0 1 {|Bs−x|≤ε} ds. Then, L 2 1 (x) is the almost sure limit of J ε (x) 2 as ε → 0 so that and by Fubini's theorem for n ∈ N even, From the occupation times formula, for every x ∈ R, for every ε > 0, So, for every x ∈ R, for every ε > 0, J ε (x) 2 B n 1 is dominated by (L * 1 ) 2 B n 1 which belongs to L 1 since L * 1 and B 1 have moments of any order (see [7] for instance). By dominated convergence theorem, we get But, when (p t ) t is the Markov transition kernel of the Brownian motion B, (1 − t + s) n/2 2π(t − s) dsdt. ⋄ To get the non-independence, one computes then for n even leading to the non-independence of ∆ 1 and B 1 . ⋄.

Conclusions and open questions
The functional limit theorem we have proved here, with an horizontal component normalized in n 3/4 with a non-Gaussian behavior and a more standard vertical one normalized in n 1/2 , strongly indicates the possibility of a local limit theorem where the probability for the walk to come back at the origin would be of order n 5/4 , in complete coherence with the transience result of [3]. This result is not straightforward and an extra work is needed to get it; this is a work in progress. Other interesting questions could concern different lattices, with e.g. also vertical orientations, but the peculiarities of these studies on oriented lattices is that the methods used are not robust for the moment. Getting robust methods for general oriented lattices seems to be a nice challenge.