Stable Limit Theorem for U-Statistic Processes Indexed by a Random Walk

Let (S_n)_{n\in\N} be a Z-valued random walk with increments from the domain of attraction of some \alpha-stable law and let (\xi(i))_{i\in\Z} be a sequence of iid random variables. We want to investigate U-statistics indexed by the random walk S_n, that is U_n:=\sum_{1\leq i<j\leq n}h(\xi(S_i),\xi(S_j)) for some symmetric bivariate function h. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the U-statistic U_n.


Introduction
Random walks in random scenery were introduced by Kesten and Spitzer [13] and lead to a class of self-similar processes with scaling not equal to √ n, which is the scaling for sums of independent or short range dependent random variables. We want to study not partial sums, but U -statistics indexed by a stable random walk, which is defined as follows: Let (X n ) n∈N be an iid (independent identically distributed) sequence of Z-valued random variables with EX 1 = 0 and the property that the law L(X 1 ) is in the normal domain of attraction of an α-stable law F α with 0 < α ≤ 2, i.e.: for S n := n m=1 X m one has P (n − 1 α S n ≤ x) → F α (x). It is then well known that the sequence of stochastic processes S (n) t := n − 1 α S [nt] ; t ≥ 0, n ∈ N converges in distribution towards an α-stable Lévy process S ⋆ t (see Skorokhod [19], Theorem 2.7). We assume that the random walk (S n ) n∈N is irreducible and strongly aperiodic. Moreover, let (ξ(i)) i∈Z be a sequence of iid random variables. When α > 1, Kesten and Spitzer [13] studied the partial sum process n i=1 ξ(S i ) and showed that it converges weakly to a self-similar process (see also [2,9,4] for the case when α ≤ 1). Instead of partial sums, we want to investigate the asymptotic behaviour of U -statistics, which can be regarded as generalized partial sums. Let h be a bivariate, measurable and symmetric function, such that A classic approach for U -statistics in the case of finite second moments is the Hoeffding decomposition [12]. Without loss of generality, we can assume that Eh(ξ(1), ξ(2)) = 0 and we can write We call L n := n i=1 h 1 (ξ(S i )) the linear part of the U -statistic U n and R n = 1≤i<j≤n h 2 (ξ(S i ), ξ(S j )) the remainder term. Note that under second moments, the summands of the remainder R n are uncorrelated.
One might expect the linear part to dominate the asymptotic behaviour (and we will indeed show this). But this is not obvious, as the random walk in random scenery shows some long range dependent behaviour. For other models of long range dependence (e.g. Gaussian sequences with slowly decaying covariances), both the linear part and the remainder term might contribute to the limit distribution. Because of this, there are other methods like representing the U -statistics as a functional of the empirical distribution function, see Dehling and Taqqu [8] or Beutner and Zähle [1]. Furthermore, the Hoeffding decomposition uses the fact that the summands of the remainder term are uncorrelated and thus it requires second moments in its original form.
Using the Hoeffding decomposition and truncation arguments, Heinrich and Wolf studied U -statistics without finite second moments. An alternative approach using point processes was used by Dabrowski, Dehling, Mikosch, Sharipov [7]. The Ustatistic indexed by a random walk was examined by Cabus and Guillotin-Plantard [3] and Guillotin-Plantard and Ladret [10], but only in the case of finite fourth moments. Our first aim is to show the convergence of the U -statistic process indexed by S n towards stable, long-range dependent, self-similar processes in the case that this moment condition does not hold.
Furthermore, we want to establish a law of the iterated logarithm for the Ustatistic process indexed by S n , extending results from Lewis [16] and Zhang [20] for the partial sum indexed by a random walk.

Main Results
Our first theorem will establish the weak convergence of the U -statistic process not assuming that the summands of the linear part have second (or even higher) moments. More precisely, we will assume that the law L(h 1 (ξ(1))) is in the normal domain of attraction of an β-stable law F β with 1 < β ≤ 2 For 1 < α ≤ 2 or β = 2, the random walk in random scenery converges to a smooth limit (see [13,2,9]), even if the scenery contains jumps, so we define the continuous version of the U -statistics process U n (t) = U k if nt = k ∈ N and linear interpolated in between. We will prove weak convergence in the space of continuous functions C[0, 1] equipped with the supremum norm. For α ≤ 1 and β < 2, the limit process has a discontinuous limit (see Castell, Guillotin-Plantard, Pène [4]), so we will consider the space of càdlàg-functions D[0, 1] endowed with the Skorohod M 1 -topology (see Skorokhod [18]).
with ∆ t as defined in Kesten and Spitzer [13]. If α = 1 and if 1 n n i=1 X i converges in distribution to aZ for a Cauchy-distributed random variable Z, we have the weak convergence in D[0, 1] Observe that we can choose β ′ such that η < β, that means that the summands h of the U -statistic might have less moments than h 1 , so without loss of generality, we can assume that E|h 1 (ξ(1))| η < ∞.
To give the definition of the process (∆ t ) t∈[0,1] , we have to introduce some notation. Let (T t (x)) t≥0 be the local time of the limit process (S ⋆ t ) t≥0 of the rescaled partial sum (n − 1 almost surely. Let (Z + (t)) t≥0 and (Z − (t)) t≥0 be to independent copies of the limit process of the rescaled partial sum process n − 1 . Then the limit process of the random walk in random scenery is defined as For random walks in random scenery, Lewis [16] and Khoshnevisan and Lewis [14] proved the law of the iterated logarithm. This was improved by Csáki, König, Shi [6] and Zhang [20] using strong approximation methods. In our second theorem, we will extend these results to U -statistics: Theorem 2. Let the assumption of Theorem 1 hold with α = β = 2 and additional E|h 1 (ξ(i))| p < ∞ and E|X i | p < ∞ for some p > 2. Then 3 Var(X) 1 4 almost surely.

Auxiliary Results
We define the occupation times N n (x) : is n plus the number of self intersections of the random walk.
Proof. We define a l = 2 l 1+β ′ α ′ β ′ and the truncated kernel We also need the Hoeffding decomposition for the truncated kernel: We introduce the following notation: Recall the Hoeffding decomposition Similar, we have thatŨ l,n = (n − 1)L l,n +R l,n . We now obtain the following representation for the remainder term: We will treat the three summands separately, so we have to show that (1) With a.s. we indicate that the convergence holds almost surely. In the proof of (1), we have to deal with the problem that we might have S i = S j ) for i = j, so we will treat these two cases separately: In order to establish bounds for the maximum, we have to control the increments of A l,n . Let n 1 , n 2 ∈ N with n 1 ≤ n 2 ≤ 2 l , then Si =Sj (h(ξ(S i ), ξ(S j )) − h 0,l (ξ(S i ), ξ(S j ))), so we have at most 2 l (n 2 − n 1 ) summands of A l,n2 − A l,n1 and for every summand Consequently, we have by the triangular inequality that We can write A l,n = n i=1 (A l,i − A l,i−1 ) (with A l,0 := 0) and in the same way A l,n2 − A l,n1 = n2 i=n1+1 (A l,i − A l,i−1 ), so we can apply the maximal inequality in Theorem 3 of Móricz [17] and obtain With the Borel-Cantelli Lemma, we can now conclude that A l,n ≥ ǫ infinitely often = 0 and thus max k≤2 l A l,n = o a.s. (2 ). For B l,n , we use the fact that the sequences (S n ) n∈N and (ξ(n)) n∈N are independent and observe that for some constant C, where we used Proposition 3.1 for the ocupation times N n (x) := n i=1 1 {Si=x} . Again using the Markov inequality, we arrive at α ′ β l l < ∞ and the Borel-Cantelli Lemma leads to max k≤2 l B l,n = o a.s. (2 l(2− 1 α ′ + 1 α ′ β ) ) as above, which completes the proof of (1). To prove (2), note that With the triangular inequality and the assumption that E|h(ξ(1), ξ(2))| η < ∞, it follows that for some constant C and any n 1 , n 2 ∈ N with n 1 ≤ n 2 Again, we apply the maximal inequality in Theorem 3 of Móricz [17] and obtain E max n≤2 l 2 l |L n −L l,n | ≤ C2 2l a 1−η l l for some constant C and we can proceed in the same way as we proved almost sure convergence as for A l,n . So it remains to show the last part (3). We will prove that We obtain with a short calculation that R l,n =Ũ l,n − (n − 1)L l,n = (Ũ l,n − U n ) + (n − 1)(L n −L l,n ) + R n and consequently We have already shown in (1) and (2) that the first two summands are of order o(2 ). For the last summand, we use the fact that Eh 2 (ξ(1), ξ(2)) = Eh(ξ(1), ξ(2)) − Eh 1 (ξ(1)) − Eh 1 (ξ(2)) = 0 and that E|h 2,l (ξ(1), ξ(1))| ≤ E|h 0,l (ξ(1), ξ(1))| + 2E|h 1,l (ξ(1))| ≤ E|h(ξ(1), ξ(1))| + 2E|h(ξ(1), ξ(2))| < ∞, so only the indices with S i = S j contribute to the expectation Si=Sj with Proposition 3.1. To show the convergence of the remaining part, we first decompose it as (1), ξ(1))) =: C l,n + D l,n .
For D l,n , we have by the independence of (S n ) n∈N and (ξ(n)) n∈N as E|h 2,l (ξ(1), ξ(1))| < ∞ and in the same way as for B l,n we can conclude that max k≤2 l D l,n = o a.s. (2 ). Finally, we will deal with C l,n . Recall that h 0,l is bounded by a l , so h 2,l is bounded by 3a l . By the triangular inequality for the L η -norm, we have that and as a consequence for some constant C > 0 E (h 2,l (ξ(1), ξ(2))) 2 ≤ (3a l ) 2−η E|h 2,l (ξ(1), ξ(2))| η ≤ C2 Furthermore we have the property of the Hoeffding-decomposition that the random variables h 2,l (ξ(1), ξ(2)) and h 2,l (ξ(1), ξ(3)) are uncorrelated, see Lee [15], page 30. So we can find bounds for the conditional variance of the increments of C l,n . To simplify the notation, we write Y (i, j) := h 2,l (ξ(i), ξ(j))1 {i =j} − (Eh 2,l (ξ(i), ξ(j))) 1 {i =j} and obtain for n 1 ≤ n 2 ≤ 2 l So the conditions of Theorem 3 of Móricz [17] hold for the (random) superadditive function It follows that Taking the expectation with respect to (X k ) k∈N , we get the following bound using Proposition 3.1 at We can now use the Chebyshev inequality and arrive at and the Borel-Cantelli Lemma completes the proof.
For the remainder term, we have proved in Proposition 3.2 that in probability. The statement of the theorem follows by Slutzky's theorem. In the cases α ≤ 1, note that the uniform convergence of the remainder R [nt] implies the convergence with respect to the M 1 -topology. The convergence of the linear part is stated in Theorem 1 and Remark 2 of Castell, Guillotin-Plantard, Pène [4]. 3 Var(X) 1 4 almost surely, which leads to the statement of the Theorem.