ELECTRONICCOMMUNICATIONSinPROBABILITY A NON-BALLISTIC LAW OF LARGE NUMBERS FOR RANDOM WALKS IN I.I.D. RANDOM ENVIRONMENT

We prove that random walks in i.i.d. random environments which oscillate in a given direction have velocity zero with respect to that direction. This complements existing results thus giving a general law of large numbers under the only assumption of a certain zero-one law, which is known to hold if the dimension is two.

Then, there exist deterministic v , v − ∈ R d (possibly zero) such that Remark: The version of this result which is quoted in [3, Theorem 3.2.2] assumes for some other purpose that the environment is uniformly elliptic, i.e. that P-a.s. ω(z, z + e) > ε for some uniform "ellipticity constant" ε > 0. However, an inspection of the proof of this theorem shows that it actually does not use this condition.
So far there is no general law of large numbers under the assumption of (1) only which would state the existence of a deterministic v towards which X n /n converges P 0 -a.s.. There are two problems to be solved in order to derive such a law from Theorem A: (i) Show that v = 0 if 0 < P 0 [A ] < 1. Or even show that 0 < P 0 [A ] < 1 is impossible. The latter has been proven in [4] for d = 2 and is still unknown for d ≥ 3. (ii) Show that for the elements of some basis of Z d the assumption Of course, the main statement here is that the limit in (2) P 0 -a.s. exists. Once this has been established it follows from P 0 [A e ∪ A −e ] = 0 that the value of this limit has to be 0.

Proofs
We first introduce some more notation. Fix some standard basis vector e ∈ Z d , |e| = 1. The following quantities are functions of nearest neighbor paths X · = (X n ) n . For any 0 ≤ u ∈ R, denote by T u := inf{n ≥ 0 | X n e ≥ u} (≤ ∞) the first time the e-coordinate of X · reaches or exceeds the level u. The times spent by X · inside the hyperplane at distance m from the origin before X · enters the hyperplane at distance m + L constitute the set T m,L := {n ≥ 0 | T m ≤ n < T m+L , X n e = m} (m, L ∈ N).
Note that T m ∈ T m,L if T m,L = ∅. The diameter of T m,L is denoted by h m,L . Finally, some empirical cumulative distribution function related to h m,L is given by Notice that T m,L and h m,L increase in L as sets and numbers, respectively, whereas F M,L (c) decreases in L and increases in c. The following lemma deals with properties of single paths.
Lemma 3. Let (X n ) n ∈ (Z d ) N be a fixed nearest neighbor path with X 0 = 0 and lim sup n→∞ X n e/n > 0. Then Proof. By assumption there exist δ > 0 and a strictly increasing sequence (n k ) k of positive integers such that Define M k := (1 − δ/2) n k δ/2 . We are going to show that for all L ∈ N, for all k large enough. To this end, fix L. Since X · is a nearest neighbor path it follows from Applying this to α = (1 − δ/2)n k + 2L/δ shows that for k large enough. For the proof of (5) observe that the sets T m,L (m = 0, . . . , M k ) are disjoint subsets of N and that all their elements are strictly less than T M k +L . Consequently, the left-hand side of (5) is at most T M k +L which along with (7) yields (5). For the proof of (6) note that h m,L ≤ T m+L − T m for all m. Hence the left-hand side of (6) is at most from which (6) follows again by (7). Now assume that (3) is false. Then we define recursively a strictly increasing sequence (L i ) i≥0 as follows. Set L 0 := 0, suppose that L i has already been defined, and set c i : Then for any i, We split the above sum canonically into three sums and get from (6) and the definition of c i that the first sum is less than 1/3 for large k. Due to (8), the second sum is less than 1/3 for large k as well. Hence for any i, for large enough k. Now set i 0 := 12/δ . Since (L i ) i is an increasing sequence, (T m,Li ) i is an increasing sequence of sets for all m. Hence #T m,Li < #T m,Li+1 if h m,Li < h m,Li+1 . Therefore, (5) with L = L i0 yields for large k due to (9), which is a contradiction.
Proof of Theorem 1. The proof is by contradiction. Assume that Since lim sup M →∞ F M,L (c) is measurable, increasing in c, and decreasing in L it follows from (10) and Lemma 3 that for some finite c, which will be kept fixed for the rest of the proof. Now we need some more notation. Let H 1 (x) := inf{n ≥ 0 | X n = x} be the first-passage time of the walk time through x (x ∈ Z d ) and H r (x) := inf{n > H r−1 (x) | X n = x} be the time of the r-th visit to x (r ≥ 2). Consider the last point visited by the walker in the hyperplane at distance m before the walker reaches the hyperplane at distance m + L. On the event {h m,L ≤ c}, this point lies within | · | 1 -distance c from X Tm and has been visited at most c times before T m+L . This means on {h m,L ≤ c} there are z ∈ Z d with |z| 1 ≤ c, ze = 0 and r ∈ N with 1 ≤ r ≤ c such that the event Since there are only finitely many such z and r it follows from (11) that for some of them  The hyperplane at distance m + L is reached before x + z has been visited twice. Hence B 1 m,L (z, r = 2) does not occur. However, the auxiliary walk Y x+z · lets B 2 m,L happen by entering the hyperplane at distance m + L before visiting the hyperplane at distance m for a second time.
We keep a pair of z and r with property (12) fixed and drop z and r from the notation. For fixed L, the sequence (1 B 1 m,L ) m seems to have a complicated dependence structure under P 0 . However, one can dominate this sequence by an auxiliary sequence (1 Bm,L ) m , which consists of L i.i.d. sequences. To this end, we create for given ω in addition to the RWRE X · , for each starting point y ∈ Z d an additional RWRE Y y · such that X · and all Y y · (y ∈ Z d ) are independent of each other, given ω. More precisely, we consider the probability measure endowed with its canonical σ-algebra and then realize X · and the Y y · 's as projections. Again, P 0 := E × P 0,ω . We then define Here and in the following, if a stopping time is applied to a path other than X · , then this path is added in parentheses after the symbol for the stopping time.
If we interpret B 1 m,L like B 2 m,L as an event in the big sample space given in (13), we have B 1 m,L ⊆ B m,L and hence by (12), We shall show at the end of the proof that for any 0 ≤ i < L, the events Assuming this, we get from the ordinary strong law of large numbers and (14) that Here the last inequality follows from [4,Lemma 4], which implies that it is P 0 -a.s. impossible for the walker to visit the strip {y ∈ Z d | 0 ≤ ye ≤ u} (u ≥ 0) infinitely often without ever visiting the half space {y | ye < 0} to the left of the strip. However, P 0 [A e ] > 0 contradicts the assumption P 0 [A e ∪ A −e ] = 0. Hence (10) is false. Repeating the argument with e in (10) replaced by −e proves (2). It remains to show (15). Let k ≥ 0 and 0 ≤ m 0 < . . . < m k with m j mod L = i for all j = 0, . . . , k. As above it follows from [4, Lemma 4] and P 0 [A −e ] = 0 that T m k is P 0 -a.s. finite because otherwise the walker could visit a strip of finite width infinitely often without visiting the half space to the right of the strip. Therefore, since B 1 m,L and B 2 m,L are disjoint, Here we used in (16) the strong Markov property with respect to σ m k and in (16) and (17) the fact that P 0,ω is a product measure, see (13). However, Consequently, the whole sum in (16) and (17) can be rewritten as x E P 0,ω B m0,L ∩ . . . ∩ B m k−1 ,L , X Tm k = x P x+z,ω [D * ≥ T m k +L ] .
Here the reason for introducing the auxiliary RWREs Y y · becomes clear: We got rid of the event {σ m k < T m k +L }, which links the environment to the left of m k with the environment in the strip of width L to the right of m k . Now the P 0,ω term in (18) is σ(ω(y, ·) | ye < m k )measurable since m j + L ≤ m k for all j < k whereas the P x+z,ω term is σ(ω(y, ·) | ye ≥ m k )measurable. Hence by independence and translation invariance (18) equals From this (15) follows by induction over k.