Let (Sn)n2N be a random walk in the domain of attraction of
an a -stable Lévy process and ( (n))n2N a sequence of iid random variables
(called scenery). We want to investigate U-statistics indexed by the random
walk Sn, that is Un :=
P
1 i<j n h( (Si);  (Sj )) for some symmetric bivariate
function h. We will prove the weak convergence without the 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 Un

Franke, Brice

Wendler, Martin

Eldorado - Repository of the TU Dortmund

SFB 823 Stable limit theorem for U-statistic processes indexed by a random walk Discussion Paper  Brice Franke, Martin Wendler      Nr. 57/2012        STABLE LIMIT THEOREM FOR U-STATISTIC PROCESSESINDEXED BY A RANDOM WALKBRICE FRANKE AND MARTIN WENDLERAbstract. Let (Sn)n∈N be a random walk in the domain of attraction ofan α-stable Lévy process and (ξ(n))n∈N a sequence of iid random variables(called scenery). We want to investigate U -statistics indexed by the randomwalk Sn, that is Un :=∑1≤i<j≤n h(ξ(Si), ξ(Sj)) for some symmetric bivariatefunction h. We will prove the weak convergence without the assumption ofﬁnite variance. Additionally, under the assumption of ﬁnite moments of ordergreater than two, we will establish a law of the iterated logarithm for theU -statistic Un.1. IntroductionRandom walks in random scenery were introduced by Kesten and Spitzer [9] andlead to a class of self-similar processes with scaling not equal to√n, which is thescaling for sums of independent or short range dependent random variables. Wewant to study not partial sums, but U -statistics indexed by a stable random walk,which is deﬁned as follows:Let (Xn)n∈N be an iid (independent identically distributed) sequence of Z-valuedrandom variables with EX1 = 0 and the property that the law L(X1) is in thedomain of attraction of an α-stable law Fα with 1 < α ≤ 2, i.e.: for Sn :=∑nm=1Xmone hasP (n−1αSn ≤ x)→ Fα(x).It is then well known that the sequence of stochastic processesS(n)t := n− 1αS[nt]; t ≥ 0, n ∈ Nconverges in distribution towards an α-stable Lévy process S?t (see Skorokhod [14],Theorem 2.7). Moreover, let (ξ(i))i∈Z be a sequence of iid random variables. Kestenand Spitzer [9] studied the partial sum process∑ni=1 ξ(Si) and showed that itconverges weakly to a self similar process. Instead of partial sums, we want toinvestigate the asymptotic behaviour of U -statistics, which can be regarded alsgeneralized partial sums. Let h be a bivariate, measurable and symmetric function,such thatE[|h(ξ(1), ξ(2))|] <∞ and E[|h(ξ(1), ξ(1))|] <∞.The U -statistic indexed by the stable random walk (Sn)n∈N is the given byUn :=∑1≤i<j≤nh(ξ(Si), ξ(Sj)).Key words and phrases. random walk; random scenery; U -statistics; stabel limits; law of theiterated logarithm.12 B. FRANKE, M. WENDLER,A classic approach for U -statistics in the case of ﬁnite second moments is the Hoeﬀ-ding decomposition [8]. Without loss of generality, we can assume that Eh(ξ(1), ξ(2)) =0 and we can writeUn = (n− 1)n∑i=1h1(ξ(Si)) +∑1≤i<j≤nh2(ξ(Si), ξ(Sj))withh1(x) := E(h(x, ξ(1)))h2(x, y) := h(x, y)− h1(x)− h1(y).We call Ln :=∑ni=1 h1(ξ(Si)) the linear part of the U -statistic Un and Rn =∑1≤i<j≤n h2(ξ(Si), ξ(Sj)) the remainder term. Note that under second moments,the summands of the remainder Rn are uncorrelated.One might expect the linear part to dominate the asymptotic behaviour (andwe will indeed show this). But this is not obvious, as the random walk in randomscenery shows some long range dependent behaviour. For other models of longrange dependence (e.g. Gaussian sequences with slowly decaying covariances), boththe 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 afunctional of the empirical distribution function, see Dehling and Taqqu [5] orBeutner and Zähle [1]. Furthermore, the Hoeﬀding decomposition uses the factthe summands of the remainder term are uncorrelated and thus it requires secondmoments in its original form.Using the Hoeﬀding decomposition and truncation arguments, Heinrich and Wolfstudied U -statistics without ﬁnite second moments. An alternative approach usingpoint processes was used by Dabrowski et al. [4]. The U -statistic indexed by arandom walk was examined by Cabus and Guillotin-Plantard [2] and Guillotin-Plantard and Ladret [6], but only in the case of ﬁnite fourth moments. Our ﬁrstaim is to show the convergence of the U -statistic process indexed by Sn towardsstable, long-range dependent, selfsimilar processes in the case that this momentcondition does not hold.Furthermore, we want to establish a law of the iterated logarithm for the U -statistic process indexed by Sn, extending results from Lewis [12] and Zhang [15]for the partial sum indexed by a random walk.2. Main ResultsOur ﬁrst theorem will establish the weak convergence of the U -statistic processnot assuming that the summands of the linear part have second (or even higher)moments:Theorem 1. Assume that the law L(X1) is in the domain of attraction of an α-stable law Fα with 1 < α ≤ 2 and that the law L(h1(ξ(1))) is in the domain of attrac-tion of an β-stable law Fβ with 1 < β ≤ 2. Furthermore, let E|h(ξ(1), ξ(2))|η <∞with η = 2β′1+β′ for a β′ > β. Then we have the weak convergence(n−2+1α− 1αβU[nt])t∈[0,1]→ (∆t)t∈[0,1],with ∆t as deﬁned in Kesten and Spitzer [9].U-STATISTIC PROCESSES INDEXED BY RANDOM WALK 3It is possible that η < β, that means that the summands h of the U -statisticmight have less moments than h1. Without loss of generality, we can assume thatη < β, so E|h1(ξ(1))|η <∞.To give the deﬁnition of the process (∆t)t∈[0,1], we have to introduce some no-tation. Let (Tt(x))t≥0 be the local time of the limit process (S?t )t≥0 of the rescaledpartial sum (n−1α∑[nt]i=1Xi)t≥0, that means∫ t01[a,b)(S?s )ds =∫ baTt(x)dxalmost surely. Let (Z+(t))t≥0 and (Z−(t))t≥0 be to independent copies of the limitprocess of the rescaled partial sum process(n−1β∑[nt]i=1 h1(ξ(i)))t≥0. Then the limitprocess of the random walk in random scenery is deﬁned as∆t =∫ ∞0Tt(x)dZ+(x) +∫ ∞0Tt(−x)dZ−(x).For random walks in random scenery, Lewis [12] and Khoshnevisan and Lewis[10] proved the law of the iterated logarithm. This was improved by Csáki et al.[3] and Zhang [15] 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 additionalE|h1(ξ(i))|p <∞ and E|Xi|p <∞ for some p > 2. Thenlim supn→∞Unn74 (log log n)34=214 Var(ξ)123 Var(X)14lim infn→∞Unn74 (log log n)34= −214 Var(ξ)123 Var(X)14almost surely.3. Auxiliary ResultsWe deﬁne the occupation times Nn(x) :=∑ni=1 1{Si=x}.Lemma 3.1. For k, p ≥ 1E(∑x∈ZNpn(x))k= O(nkp(1−1α )+k1α )This is Lemma 2.1 of Guillotin-Plantard, Ladret [6]Proposition 3.2. Under the conditions of Theorem 1, we hvae thatmaxk≤nRk = o(n1− 1α+ 1αβ )almost surely.Proof. We deﬁne al = 2l 1+β′αβ′ and the truncated kernelh0,l(x, y) :={h(x, y) if |h(x, y)| ≤ al0 else.4 B. FRANKE, M. WENDLER,We also need the Hoeﬀding decomposition for the truncated kernel:h1,l(x) := E(h0,l(x, ξ(1)))h2,l(x, y) := h0,l(x, y)− h1,l(x)− h1,l(y).We introduce the following notation:L˜l,n :=n∑i=1h1,l(ξ(Si))U˜l,n :=∑1≤i<j≤nh0,l(ξ(Si), ξ(Sj))R˜l,n :=∑1≤i<j≤nh2,l(ξ(Si), ξ(Sj))Recall the Hoeﬀding decompositionUn = (n− 1)Ln +Rn.Similar, we have thatU˜l,n = (n− 1)L˜l,n + R˜l,n.We now obtain the following representation for the remainder term:Rn = Un − (n− 1)Ln = (Un − U˜l,n)− (n− 1)Ln + U˜l,n= (Un−U˜l,n)−(n−1)Ln+(n−1)L˜l,n+R˜l,n = (Un−U˜l,n)−(n−1)(Ln−L˜l,n)+R˜l,nWe will treat the three summands separately, so we have to show that(1)maxn≤2l|Un − U˜l,n| = oa.s.(2l(2− 1α+ 1αβ )),(2)maxn≤2l2l|Ln − L˜l,n| = oa.s.(2l(1− 1α+ 1αβ )),(3)maxn≤2l|R˜l,n| = oa.s.(2l(2− 1α+ 1αβ )).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 6= j, so wewill treat these two cases separately:|Un − U˜l,n| ≤∣∣∣∣∣∣∣∣∑1≤i<j≤nS(i) 6=S(j)(h(ξ(Si), ξ(Sj))− h0,l(ξ(Si), ξ(Sj)))∣∣∣∣∣∣∣∣+∣∣∣∣∣∣∣∣∑1≤i<j≤nS(i)=S(j)(h(ξ(Si), ξ(Sj))− h0,l(ξ(Si), ξ(Sj)))∣∣∣∣∣∣∣∣ = |Al,n|+ |Bl,n|U-STATISTIC PROCESSES INDEXED BY RANDOM WALK 5In order to establish bounds for the maximum, we have to control the incrementsof Al,n. Let n1, n2 ∈ N with n1 ≤ n2 ≤ 2l, thenAl,n2 −Al,n1 =∑1≤i<j≤nn1<j≤n2S(i)6=S(j)(h(ξ(Si), ξ(Sj))− h0,l(ξ(Si), ξ(Sj))),so we have at most 2l(n2 − n1) summands of Al,n2 −Al,n1 and for every summandE |h(ξ(Si), ξ(Sj))− h0,l(ξ(Si), ξ(Sj))| = E|h(ξ(1), ξ(2))1{|h(ξ(1),ξ(2))|>al}|≤ a1−ηl E|h(ξ(1), ξ(2))1{|h(ξ(1),ξ(2))|>al}|η ≤ a1−ηl E|h(ξ(1), ξ(2))|η.Consequently, we have by the triangular inequality thatE|Al,n2 −Al,n1 | ≤ 2l(n2 − n1)E|h(ξ(1), ξ(2))1{|h(ξ(1),ξ(2))|>al}|≤ 2l(n2 − n1)a1−ηl E|h(ξ(1), ξ(2))|η.We can write Al,n =∑ni=1(Ai,n − Ai−1,n) (with Al,0 := 0) and in the same wayAl,n2 − Al,n1 =∑n2i=n1+1(Ai,n − Ai−1,n), so we can apply the maximal inequalityin Theorem 3 of Móricz [13] and obtainE∣∣∣∣maxk≤2lAl,n∣∣∣∣ ≤ C22la1−ηl lfor some constant C. Recall that al = 2l 1+β′αβ′ and η = 2β′1+β′ , so 1 − η = 1−β′1+β′ . Itfollows from the Markov inequality that∞∑l=1P(12l(2−1α+1αβ )maxk≤2lAl,n ≥ )≤ C∞∑l=122la1−ηl l2l(2−1α+1αβ )=C∞∑l=12l 1+β′αβ′1−β′1+β′ l2l(−1α+1αβ )=C∞∑l=12l( 1αβ′− 1α )l2l(−1α+1αβ )=C∞∑l=12l( 1αβ′− 1αβ )l <∞,as 1αβ′ − 1αβ < 0. With the Borel-Cantelli Lemma, we can now conclude thatP(12l(2−1α+1αβ )maxk≤2lAl,n ≥  inﬁnitely often)= 0and thus maxk≤2l Al,n = oa.s.(2l(2− 1α+ 1αβ )). For Bl,n, we use the fact that thesequences (Sn)n∈N and (ξ(n))n∈N are independent observe thatE|maxn≤2lBl,n| ≤ Emaxn≤2l∑1≤i<j≤nS(i)=S(j)|(h(ξ(Si), ξ(Sj))− h0,l(ξ(Si), ξ(Sj)))|≤ E∑1≤i<j≤2lS(i)=S(j)|(h(ξ(Si), ξ(Sj))− h0,l(ξ(Si), ξ(Sj)))|≤ E#{1 ≤ i < j ≤ 2l|S(i) = S(j)}E|h(ξ(1), ξ(1))1{|h(ξ(1),ξ(1))|>al}|≤ E[∑x∈ZN22l(x)]E|h(ξ(1), ξ(1))| ≤ C2l(2−1α )6 B. FRANKE, M. WENDLER,for some constant C, where we used Lemma 2.1 of Guillotin-Plantard and Ladret [6]for the ocupation timesNn(x) :=∑ni=1 1{Si=x}. Again using the Markov inequality,we arrive at∞∑l=1P(12l(2−1α+1αβ )maxk≤2lBl,n ≥ )≤ C∞∑l=12l(2−1α )2l(2−1α+1αβ )≤ C∞∑l=12−1αβ l <∞and the Borel-Cantelli Lemma leads to maxk≤2l Bl,n = oa.s.(2l(2− 1α+ 1αβ )) as above,which completes the proof of (1). To prove (2), note thatE|h1(ξ(1))−h1,l(ξ(1))| = E|E[h(ξ(1), ξ(2))|ξ(1)]−E[h(ξ(1), ξ(2))1{|h(ξ(1),ξ(2))|≤al}|ξ(1)]|= E|E[h(ξ(1), ξ(2))1{|h(ξ(1),ξ(2))|>al}|ξ(1)]|≤ E [E [|h(ξ(1), ξ(2))1{|h(ξ(1),ξ(2))|>al}|∣∣ξ(1)]]= E∣∣h(ξ(1), ξ(2))1{|h(ξ(1),ξ(2))|>al}∣∣ ≤ 1aη−1lE|h(ξ(1), ξ(2))|η.With the triangular inequality and the assumption that E|h(ξ(1), ξ(2))|η < ∞, itfollows that for some constant C and any n1, n2 ∈ N with n1 ≤ n2E|n2∑i=n1+1(h1(ξ(Si))− h1,l(ξ(Si)))| ≤ C(n2 − n1)a1−ηl .Again, we apply the maximal inequality in Theorem 3 of Móricz [13] and obtainEmaxn≤2l2l|Ln − L˜l,n| ≤ C22la1−ηl lfor some constant C and we can proceed in the same way as we proved almost sureconvergence as for Al,n. So it remains to show the last part (3). We will proof thatmaxn≤2l|ER˜l,n| = o(2l(2− 1α+ 1αβ ))maxn≤2l|R˜l,n − ER˜l,n| = oa.s.(2l(2− 1α+ 1αβ )).We obtain with a short calculation thatR˜l,n = U˜l,n − (n− 1)L˜l,n = (U˜l,n − Un) + (n− 1)(Ln − L˜l,n) +Rnand consequentlymaxn≤2l|ER˜l,n| = maxn≤2l∣∣∣E(U˜l,n − Un) + (n− 1)E(Ln − L˜l,n) + ERn∣∣∣≤ Emaxn≤2l|Un − U˜l,n|+ (n− 1)Emaxn≤2l|Ln − L˜l,n|+ maxn≤2l|ERn|.We have already shown in (1) and (2) that the ﬁrst two summands are of or-der o(2l(2−1α+1αβ )). For the last summand, we use the fact that Eh2(ξ(1), ξ(2)) =Eh(ξ(1), ξ(2))−Eh1(ξ(1))−Eh1(ξ(2)) = 0 and that E|h2,l(ξ(1), ξ(1))| ≤ E|h0,l(ξ(1), ξ(1))|+2E|h1,l(ξ(1))| ≤ 3E|h(ξ(1), ξ(1))| <∞, soERn = E ∑1≤i<j≤nS(i)=S(j)h2(ξ(Si), ξ(Sj))U-STATISTIC PROCESSES INDEXED BY RANDOM WALK 7andmaxn≤2l|ERn| ≤ maxn≤2lE# {1 ≤ i < j ≤ n|S(i) = S(j)}E|h2(ξ(1), ξ(1))|≤ E#{1 ≤ i < j ≤ 2l|S(i) = S(j)}E|h2(ξ(1), ξ(1))| ≤ C2l(2− 1α ) = o(2l(2− 1α+ 1αβ ))with Lemma 2.1 of Guillotin-Plantard and Ladret. To show the convergence of theremaining part, we ﬁrst decompose it asR˜l,n − ER˜l,n =∑1≤i<j≤nS(i)6=S(j)(h2,l(ξ(Si), ξ(Sj))− Eh2,l(ξ(1), ξ(2)))+∑1≤i<j≤nS(i)=S(j)(h2,l(ξ(Si), ξ(Sj))− Eh2,l(ξ(1), ξ(1))) =: Cl,n +Dl,n.For Dl,n, we have by the independence of (Sn)n∈N and (ξ(n))n∈NEmaxn≤2l|Dn| ≤ E#{1 ≤ i < j ≤ 2l|S(i) = S(j)}E|h2,l(ξ(1), ξ(1))| ≤ C2l(2− 1α )as E|h2,l(ξ(1), ξ(1))| < ∞ and in the same way as for Bl,n we can conclude thatmaxk≤2l Dl,n = oa.s.(2l(2− 1α+ 1αβ )). Finally, we will deal with Cl,n. Recall that h0,lis bounded by al, so h2,l is bounded by 3al. By the triangular inequality for theLη-norm, we have thatE|h2,l(ξ(1), ξ(2))|η ≤(‖h0,l(ξ(1), ξ(2))‖η + 2 ||h1,l(ξ(1))‖η)η≤(3 ‖h(ξ(1), ξ(2))‖η)η,and as a consequence for some constant C > 0E (h2,l(ξ(1), ξ(2)))2 ≤ (3al)2−ηE|h2,l(ξ(1), ξ(2))|η ≤ C2l1+β′αβ′ (2−2β′1+β′ ) = C2l 2αβ′ .Furthermore we have the property of the Hoeﬀding-decomposition that the randomvariables h2,l(ξ(1), ξ(2)) and h2,l(ξ(1), ξ(3)) are uncorrelated, see Lee [11], page 30.So we can ﬁnd bounds for the conditional variance of the increments of Cl,n. Tosimplify the notation, we writeY (i, j) := h2,l(ξ(i), ξ(j))1{i6=j} − (Eh2,l(ξ(i), ξ(j)))1{i 6=j}and obtain for n1 ≤ n2 ≤ 2lE[(Cl,n2 − Cl,n1)2∣∣(Xk)k∈N] = E ∑1≤i<j≤nn1<j≤n2Y (S(i), S(j))2 ∣∣(Xk)k∈N=∑x,y∈Zx<y(# {1 ≤ i < j ≤ n|n1 < j ≤ n2, S(i) = x, S(j) = y}+# {1 ≤ i < j ≤ n|n1 < j ≤ n2, S(i) = y, S(j) = x})2E(Y (x, y))2≤ C2l 2αβ′ 2∑x,y∈Z(Nn(x)(Nn2(y)−Nn1(y)))2.8 B. FRANKE, M. WENDLER,So the conditions of Theorem 3 of Móricz [13] hold for the (random) superadditivefunctiong(Fn1,n2) =∑x,y∈ZNn(x)2(Nn2(y)−Nn1(y))2.It follows thatEmaxn≤2l ∑1≤i<j≤nY (S(i), S(j))2 ∣∣(Xk)k∈N ≤ C2l 2αβ′ ∑x∈ZN22l(x)∑y∈ZN22l(y)l2Taking the expectation with respect to (Xk)k∈N, we get the following bound usingLemma 2.1 of Guillotin-Plantard and Ladret [6] atE(maxn≤2lCl,n)2= EEmaxn≤2l ∑1≤i<j≤nY (S(i), S(j))2 ∣∣(Xk)k∈N≤ C2l 2αβ′ l2E(∑x∈ZN22l(x))2≤ C2l 2αβ′ l22l(4− 2α ) = C2l2(2− 1α+ 1αβ′ )l2.We can now use the Chebyshev inequality and arrive at∞∑l=1P(12l(2−1α+1αβ )maxk≤2lCl,n ≥ )≤ C2∞∑l=12l2(2− 1α+ 1αβ′ )l22l2(2−1α+1αβ )≤ C2∞∑l=12l2( 1αβ′− 1αβ )l2 <∞and the Borel-Cantelli Lemma completes the proof. 4. Proofs of Main ResultsProof of Theorem 1. Recall the Hoeﬀding decompositionn−2+1α− 1αβU[nt] =n− 1nn−1+1α− 1αβL[nt] + n−2+ 1α− 1αβR[nt].For the linear part, we apply Theorem 1.1 of Kesten and Spitzer [9] to the randomvariables h1(ξ(i)) and conclude that(n−1+1α− 1αβL[nt])t∈[0,1]converges weakly to(∆t)t∈[0,1]. For the remainder term, we have proved in Proposition 3.2 thatsupt∈[0,1]|n−2+ 1α− 1αβR[nt]| → 0in probability. The statement of the theorem follows by Slutzky's theorem.Proof of Theorem 2. We use again the Hoeﬀding decompositionUnn74 (log log n)34=n− 1nLn(n log log n)34+Rnn74 (log log n)34.For the remainder term we use Proposition 3.2 with α = β = 2:Rn = oa.s.(n2− 12+ 14 ) = oa.s.(n74 (log log n)34 ).U-STATISTIC PROCESSES INDEXED BY RANDOM WALK 9As Ln =∑ni=1 h1(ξ(Si)), we can aplly Corollary 1 of Zhang [15]lim supn→∞± Ln(n log log n)34=214 Var(ξ)123 Var(X)14almost surely, which leads to the statement of the Theorem. AcknowledgementThe research was supported by the DFG Sonderforschungsbereich 823 (Collab-orative Research Center) Statistik nichtlinearer dynamischer Prozesse.References[1] E. Beutner, H. Zähle, Continuous mapping approach to the asymptotics of U- and V-statistics, preprint arXiv:1203.1112.[2] P. Cabus, N. Guillotin-Plantard, Functional limit theorems for U-statistics indexedby a random walk, Stochastic Processes and their Application 101 (2002) 143-160.[3] E. Csáki, W. König, Z. Shi, An embedding for the Kesten-Spitzer random walk in randomscenery, Stochastic Processes and their Application 82 (1999) 283-292.[4] A. Dabrowski, H. Dehling, T. Mikosch, O.Sh. Sharipov, Poisson limits for U-statistics, Stochastic Processes and their Application 99 (2002) 137-157.[5] H. Dehling, M.S. Taqqu, The empirical process of some long-range dependent sequenceswith an application to U-statistics, Ann. Statist. 17 (1989) 1767-1783.[6] N. Guillotin-Plantard, V. Ladret, Limit theorems for U-statistics indexed by a onedimensional random walk, ESAIM 9 (2005) 95-115.[7] L. Heinrich, W. Wolf, On the convergence of U-statistics with stable limit distribution,Journal of Multivariate Analysis 44 (1993) 266-278.[8] W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math.Statist. 19 (1948) 293-325.[9] H. Kesten, F. Spitzer, A limit theorem related to an new class of self similar processes,Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 50 (1979) 5-25.[10] D. Khoshnevisan, T.M. Lewis, A law of the iterated logarithm for stable processes inrandom scenery, Stochastic Processes and their Applications 74 (1998) 89-121.[11] A.J. Lee, U-Statistics: Theory and Practice, Marcel Dekker, New York (1990).[12] T.M. Lewis, A law of the iterated logarithm for random walk in random scenery withdeterministic normalizer, Journal of Theoretical Probability 6 (1993) 209-230.[13] F. Móricz, Moment inequalities and the strong law of large numbers, Zeitschrift fürWahrscheinlichkeitstheorie und verwandte Gebiete 35 (1976) 299-314.[14] A.V. Skorokhod, Limit theorems for stochastic processes with independent increments, The-ory of probability and its applications 2 (1957) 138-171.[15] L. Zhang, Strong approximation for the general Kesten-Spitzer random walk in independentrandom scenery, Science in China, series A 44 (2001) 619-630.E-mail address: Martin.Wendler@rub.de   

Stable limit theorem for U-statistic processes indexed by a random walk

https://eldorado.tu-dortmund.de/bitstream/2003/29830/1/DP_5712_SFB823_Franke_Wendler.pdf

Stable limit theorem for U-statistic processes indexed by a random walk

Abstract

Similar works

Full text

Available Versions

Eldorado - Repository of the TU Dortmund