LAWS OF THE ITERATED LOGARITHM FOR TRIPLE INTERSECTIONS OF THREE DIMENSIONAL RANDOM WALKS

Abstact : Let X = f X n ; n (cid:21) 1 g , X 0 = f X 0 n ; n (cid:21) 1 g and X 00 = f X 00 n ; n (cid:21) 1 g be three independent copies of a symmetric random walk in Z 3 with E ( j X 1 j 2 log + j X 1 j ) < 1 . In this paper we study the asymptotics of I n , the number of triple intersections up to step n of the paths of X , X 0 and X 00 as n ! 1 . Our main result is where Q denotes the covariance matrix of X 1 . A similar result holds for J n , the number of points in the triple intersection of the ranges of X , X 0 and X 00 up to step n . Abstract Let X = f X n ; n (cid:21) 1 g , X 0 = f X 0 n ; n (cid:21) 1 g and X 00 = f X 00 n ; n (cid:21) 1 g be three independent copies of a symmetric random walk in Z 3 with E ( j X 1 j 2 log + j X 1 j ) < 1 . In this paper we study the asymptotics of I n , the number of triple intersections up to step n of the paths of X , X 0 and X 00 as n ! 1 . Our main result is where Q denotes the covariance matrix of X 1 . A similar result holds for J n , the number of points in the triple intersection of the ranges of X , X 0 and X 00 up to step n .

Abstact: Let X = {X n , n ≥ 1}, X = {X n , n ≥ 1} and X = {X n , n ≥ 1} be three independent copies of a symmetric random walk in Z 3 with E(|X 1 | 2 log + |X 1 |) < ∞. In this paper we study the asymptotics of I n , the number of triple intersections up to step n of the paths of X, X and X as n → ∞. Our main result is lim sup n→∞ I n log(n) log 3 (n) = 1 π|Q| a.s.
where Q denotes the covariance matrix of X 1 . A similar result holds for J n , the number of points in the triple intersection of the ranges of X, X and X up to step n.

Introduction
Let X = {X n , n ≥ 1}, X = {X n , ≥ 1}, and X = {X n , n ≥ 1} be three independent copies of a random walk in Z 3 with zero mean and finite variance. In this paper we study the asymptotics of the number of triple intersections up to step n of the paths of X, X and X as n → ∞, both the number of 'intersection times' where X(1, n) denotes the range of X up to time n and |A| denotes the cardinality of the set A. For random walks with finite variance, dimension three is the 'critical case' for triple intersections, since I n , J n ↑ ∞ almost surely but three independent Brownian motions in R 3 do not intersect. This implies that in some sense I n , J n ↑ ∞ slowly. We also note that in dimension > 3 we have I ∞ , J ∞ < ∞ a.s.
We assume that X n is adapted, which means that X n does not live on any proper subgroup of Z 3 . In the terminology of Spitzer [9] X n is aperiodic.
We have the following two limit theorems.
As usual, log j denotes the j-fold iterated logarithm.
In the particular case of the simple random walk on Z 3 , where Q = 1 3 I, Theorem 1 states that lim sup n→∞ I n log(n) log 3 (n) = 27 π a.s. (1.4) A similar result holds for J n : Theorem 2 Assume that E(|X 1 | 2 log + |X 1 |) < ∞. Then lim sup n→∞ J n log(n) log 3 (n) = q 3 π|Q| a.s. (1.5) where q denotes the probability that X will never return to its initial point.
Le Gall [4] proved that (log n) −1 J n converges in distribution to a gamma random variable. This paper is an outgrowth of my paper with Michael Marcus [6] in which we prove analogous laws of the iterated logarithm for intersections of two symmetric random walks in Z 4 with finite third moment. In this paper we also use some of the ideas of [4] along with techniques developed in [8, 7].

Proof of Theorem 1
We use p n (x) to denote the transition function for X n . Recall As shown in [9] the random walk X n is adapted if and only if the origin is the unique element of T 3 satisfying φ(p) = 1 where φ(p) is the characteristic function of X 1 and T 3 = (−π, π] 3 is the usual three dimensional torus. We use τ to denote the number of elements in the set {p ∈ T 3 | |φ(p)| = 1}. We say that X is aperiodic if τ = 1. (In Spitzer [9] this is called strongly aperiodic). We will prove our theorems for X aperiodic, but using the ideas of section 2.4 in [5] it is then easy to show that they also hold if τ > 1. According to the local central limit theorem, P7.9 and P7.10 of [9], where q t (x) denotes the transition density for Brownian motion in R 3 and Q denotes the covariance matrix of X 1 . Then, arguing seperately in the regions n ≤ n 0 , n 0 < n ≤ |x| 2 and n > |x| 2 we have we see that taking n 0 = 1 in (2.5) gives the bound We also have Changing variables, first x = ab, y = ac, z = bc, and then x = u 2 , y = v 2 , z = w 2 we have t a=1 t b=1 t c=1 1 (bc + ac + ab) 3/2 da db dc = 1≤x,y,z≤t 2 Hence, taking n 0 large in (2.5), we have Thus the assertion of Theorem 1 can be written as We begin our proof with some moment calculations.
In view of (2.20), in order to prove our lemma it suffices to show that n!
Let Ω n be the set of (σ, π, π ) for which φ σ,π and φ σ,π are both bijections. Up to the error terms described above, we can write the sum in (2.27) as Since on∆ σ we have by (2.28) that |y φ(j) | ≥ 8|v π,j − y φ(j) |, we can then replace each occurence of v π,j in (2.33) by y φ(j) , bounding the error terms using which comes from (2.17) and Lemma 5 of the Appendix.
Thus, up to error terms described which can be incorporated into R(n, t), we can write the sum in (2.33) as n! (σ,π,π )∈Ωn (y 1 ,...,yn)∈∆σ Proceeding as above, up to the error terms described above, we can replace (2.35) by and as by the remark following Lemma 2.5 in [4], we have |Ω n | = Ψ(n), the lemma is proved. 2 We will use E v,w,z to denote expectation with respect to the random walks X, X , X where X 0 = v, X 0 = w and X 0 = z. We define We will need the following lower bound.

2.
Lemma 3 For all t ≥ 0 and x = O(log log h(t)) we have Let s n = t n − t n−1 and note that, as in (2.60) of [7], we have h(s n ) ∼ h(t n ). We also note that |I tn − I t n−1 − I sn • Θ t n−1 | ≤ I tn,tn,t n−1 + I tn,t n−1 ,tn + I t n−1 ,tn,tn (2.54) where Θ n denotes the shift on paths defined by (X i , X j , X" k )(Θ n ω) = (X n+i , X n+j , X" n+k )(ω) and I n,m,p = As in Lemma 1, we can show that for t ≥ t , and for all integers n ≥ 0 and any > 0 which, as before, leads to lim sup n→∞ I tn,tn,t n−1 4h(t n ) log log h(t n ) (2.57) = lim sup n→∞ I tn,tn,t n−1 Using this for θ large, (2.54), Levy's Borel-Cantelli lemma (see Corollary 5.29 in [1]) and the Markov property, we see that (2.53) will follow from ∞ n=1 P Xt n−1 ,X t n−1 ,X t n−1 If we apply Lemma 4 with t = s n and x = log log s n we see that (2.58) will follow from ∞ n=1 a(X t n−1 , X t n−1 , X t n−1 , s n / log n) 1 n 1− = ∞ a.s. E(a(X t n−1 , X t n−1 , X t n−1 , s n / log n)) 1 n 1− = ∞ (2.60) To see this we note that E(a(X t , X t , X t , m)) (2.61) so that E(a(X t n−1 , X t n−1 , X t n−1 , s n / log n)) (2.62) Also note that This follows fairly easily since h(t) ∼ c log(t). (For the details, in a more general setting, see the proof of Theorem 1.1 of [7], especially that part of the proof surrounding (2.82)). Furthermore, we have by Hölder's inequality Taking θ large establishes (2.60).

Proof of Theorem 3
We begin with some moment calculations. Recall and note that where π runs over the set of permutations π of {1, 2, . . . , n}.
Then we see from (3.2) that We have where as usual we set p 0 (x) = 1 {x=0} . From this we see that

Consequently we have
Now it is well known that 1 1 + u t (0) ↓ q (3.9) so that for any > 0 we can find t 0 < ∞ such that for all t ≥ t 0 and x. Hence (3.3) and (3.4) give us The proof of Theorem 3 now follows exactly along the lines of the proof of Theorem 1. 2
In [3], Lawler shows that the usual Green's function asymptotics do not necessarily hold for all mean zero finite variance random walks on Z 4 . We expect that a similar analysis would show that finite variance is not enough to guarantee (4.1). Our Lemma says that E(|X 1 | 2 log + |X 1 |) < ∞ is sufficient.
For the first two integrals on the right hand side of (4.15) we integrate by parts in the p 1 direction to obtain ) dp ) dp where we have used the fact that the boundary terms coming from the integrals over A and B cancel. (These boundary terms are easily seen to be finite). We claim that (4.21) is equal to ) dp +O(|a|/|x| 2 ).
To establish our last claim, using the fact that it suffices to show that | dp < ∞ (4.23) and B |D 1 ( e −δQ(p) Q(p) )| dp < ∞ (4.24) with bounds uniform in 0 < δ ≤ 1. (4.24) is easily seen to be bounded independently of δ ≤ 1, using ideas similar to those we used in bounding the integral in (4.17). As for (4.23), we first observe that The integral of the last term is bounded easily as before. As to the first term in (4.25), we write The integrals of the two terms in the last equality are bounded by (4.8) and (4.7) respectively. This establishes our claim that (4.21) is equal to (4.22).
To bound the integrals in (4.22) we now integrate by parts once more in the p 1 direction to obtain ) dp. Once again, the (finite) boundary terms cancel. (Actually, each boundary term is O(1/|x| 2 ).) Using the bounds (4.23) and (4.24), we find that (4.27) ) dp +O(|a|/|x| 2 ) (4.28) As in the proof of (4.16), we see that To handle the first integral in (4.28) we note that Once again it is easy to control the last three terms on the right hand side of (4.30), while for the first term we use As in (4.26), we write out the first term on the right hand side of (4.31) as Hence, we can bound The integrals are now bounded using (4.6) and (4.4) respectively.
Similarly we write out the second term on the right hand side of (4.31) as (4.34) and we can bound The integrals are now bounded using (4.5) and (4.4) respectively, completing the proof of (4.1).
To prove (4.2) we first note that e ipx e −δ|p| 2 /2 − e −n|p| 2 /2 |p| 2 /2 dp. (4.37) We note that by the mean-value theorem where we have used the fact that under our assumptions Since t −1 q t (x) is, up to a constant multiple, the transition density for Brownian motion in R 5 , which has Green's function C|x| −3 , we have |v n δ (x + a) − v n δ (x)| ≤ C|a|  Therefore, it suffices to bound as before an expression of the form (4.15) where u is replaced by u n−1 and v δ is replaced by v n δ . All bounds involving v n δ on B and C are handled as before. One must verify that in each case no (divergent) factors involving n will remain. For example, whereas in the bound for the second term on the right hand side of (4.18) we were satisfied with a bound cδ 1/2 , see (4.20), when δ is replaced with n we now argue that )| ≤ cQ 2 1 (p) Q 2 (p) ≤ c|p| 2 on C, (where Q(p) ≥ d > 0) and scaling out n now gives us a bound of n −3/2 .
The analogue of the first term on the right hand side of (4.15) is − e −nQ(p) Q(p) dp.