Weak invariance principles for sums of dependent random functions

Available online at www.sciencedirect.com Stochastic Processes and their Applications 13 (013) 385 403 www.elsevier.com/locate/spa Weak invariance principles for sums of dependent random functions István Berkes a, Lajos Horváth b,, Gregory Rice b a Institute of Statistics, Graz University of Technology, Kopernikus Gasse 4/3, A 8010 Graz, Austria b Department of Mathematics, University of Utah, Salt Lake City, UT 8411 0090, USA Received 4 February 01; received in revised form 8 September 01; accepted 1 October 01 Available online 6 October 01 Abstract Motivated by problems in functional data analysis, in this paper we prove the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure. c 01 Published by lsevier B.V. MSC: 60F17; 60F5; 6G10 Keywords: Variables in Hilbert spaces; m approximability; Weak convergence 1. Introduction Functional data analysis in many cases requires central limit theorems and invariance principles for partial sums of random functions. The case of independent summands is much studied and well understood but the theory for the dependent case is less complete. In this paper we study the important class of Bernoulli shift processes which are often used to model econometric and financial data. Let X = {X i (t)} i= be a sequence of random functions, square integrable on [0, 1], and let denote the L [0, 1] norm. To lighten the notation we use f for f (t) when it does not cause confusion. Throughout this paper we assume that X forms a sequence of Bernoulli shifts, i.e. X j (t) = g(ϵ j (t), ϵ j 1 (t),...) for some nonrandom measurable function g : S L and iid random functions ϵ j (t), Research supported by SF grant DMS 0905400. Corresponding author. -mail address: horvath@math.utah.edu (L. Horváth). 0304-4149/$ - see front matter c 01 Published by lsevier B.V. doi:10.1016/j.spa.01.10.003

386 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 and < j <, with values in a measurable space S, (1.1) ϵ j (t) = ϵ j (t, ω) is jointly measurable in (t, ω) ( < j < ), (1.) X 0 (t) = 0 for all t, and X 0 +δ < for some 0 < δ < 1, (1.3) the sequence {X n } n= can be approximated by l-dependent sequences {X n,l } n= in the sense that ( X n X n,l +δ ) 1/κ < for some κ > + δ, where X n,l is defined by X n,l = g(ϵ n, ϵ n 1,..., ϵ n l+1, ϵ n,l ), ϵ n,l = (ϵ n,l,n l, ϵ n,l,n l 1,...), where the ϵ n,l,k s are independent copies of ϵ 0, independent of {ϵ i, < i < }. (1.4) We note that assumption (1.1) implies that X n is a stationary and ergodic sequence. Hörmann and Kokoszka [19] call the processes satisfying (1.1) (1.4) L m-decomposable processes. The idea of approximating a stationary sequence with random variables which exhibit finite dependence first appeared in [1] and is used frequently in the literature (cf. [3]). Aue et al. [1] provide several examples when assumption (1.1) (1.4) hold which include autoregressive, moving average and linear processes in Hilbert spaces. Also, the non-linear functional ARCH(1) model (cf. [18]) and bilinear models (cf. [19]) satisfy (1.4). We show in Section (cf. Lemma.) that the series in C(t, s) = [X 0 (t)x 0 (s)] + [X 0 (t)x l (s)] + [X 0 (s)x l (t)] (1.5) are convergent in L. The function C(t, s) is positive definite, and therefore there exist λ 1 λ 0 and orthonormal functions φ i (t), 0 t 1 satisfying λ i φ i (t) = C(t, s)φ i (s)ds, 1 i <, (1.6) where means 1 0. We define Γ (x, t) = λ 1/ i W i (x)φ i (t), where W i are independent and identically distributed Wiener processes (standard Brownian motions). Clearly, Γ (x, t) is Gaussian. We show in Lemma. that λ l <, and therefore Γ (x, t)dt < a.s. sup 0 x 1 Theorem 1.1. If assumption (1.1) (1.4) hold, then for every we can define a Gaussian process Γ (x, t) such that {Γ (x, t), 0 x, t 1} D ={Γ (x, t), 0 x, t 1}

and where sup 0 x 1 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 387 (S (x, t) Γ (x, t)) dt = o P (1), S (x, t) = 1 x 1/ X i (t). The proof of Theorem 1.1 is given in Section. The proof is based on a maximal inequality which is given in Section 3 and is of interest in its own right. There is a wide literature on the central limit theorem for sums of random processes in abstract spaces. For limit theorems for sums of independent Banach space valued random variables we refer to Ledoux and Talagrand (1991). For the central limit theory in the context of functional data analysis we refer to the books of [4,0]. In the real valued case, the martingale approach to weak dependence was developed by Gordin [16], Philipp and Stout [5] and berlein [14], and by using such techniques [3,9] obtained central limit theorems for a large class of dependent variables in Hilbert spaces. For some early influential results on invariance for sums of mixing variables in Banach spaces we refer to [,11,10]. These papers provide very sharp results, but verifying mixing conditions is generally not easy and without additional continuity conditions, even autoregressive (1) processes may fail to be strong mixing (cf. [5]). The weak dependence concept of [13] (cf. also [8]) solves this difficulty, but so far this concept has not been extended to variables in Hilbert spaces. Wu [7,8] proved several limit theorems for one-dimensional stationary processes having a Bernoulli shift representation. Compared to classical mixing conditions, Wu s physical dependence conditions are easier to verify in concrete cases. Condition (1.3) cannot be directly compared to the approximating martingale conditions of [7,8]. For extensions to the Hilbert space case we refer to [19].. Proof of Theorem 1.1 The proof is based on three steps. We recall the definition of X i,m from (1.4). For every fixed m, the sequence {X i,m } is m-dependent. According to our first lemma, the sums of the X i s can be approximated with the sums of m-dependent variables. The second step is the approximation of the infinite dimensional X i,m s with finite dimensional variables (Lemma.4). Then the result in Theorem 1.1 is established for finite dimensional m-dependent random functions (Lemma.6). Lemma.1. If (1.1) (1.4) hold, then for all x > 0 we have 1 k lim lim sup P max (X i X i,m ) m 1 k > x = 0. (.1) Proof. The proof of this lemma requires the maximal inequality of Theorem 3.. Section 3 is devoted to the proof of this result. Using Theorem 3., (.1) is an immediate consequence of Markov s inequality. Define m m C m (t, s) = [X 0,m (t)x 0,m (s)] + [X 0,m (t)x i,m (s)] + [X 0,m (s)x i,m (t)]. (.)

388 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 We show in the following lemma that for every m the function C m is square-integrable. Hence there are λ 1,m λ,m 0 and corresponding orthonormal functions φ i,m, i = 1,,... satisfying λ i,m φ i,m (t) = C m (t, s)φ i,m (s)ds, i = 1,,... Lemma.. If (1.1) (1.4) hold, then we have C (t, s)dtds <, and (.3) Cm (t, s)dtds < for all m 1, (.4) lim m (C(t, s) C m (t, s)) dtds = 0, (.5) C(t, t)dt = λ k <, (.6) C m (t, t)dt = lim m k=1 λ k,m < (.7) k=1 C m (t, t)dt = C(t, t)dt. (.8) Proof. Using the Cauchy Schwarz inequality for expected values we get ([X 0 (t)x 0 (s)]) dtds (( X0 (t))1/ ( X0 (s))1/ ) dtds = ( X 0 ) <. Recalling that X 0 and X i,i are independent and both have 0 mean, we conclude first using the triangle inequality and then the Cauchy Schwarz inequality for expected values that 1/ [X 0 (t)x i (s)] dtds = [X 0 (t)(x i (s) X i,i (s))] dtds X 0 (t)(x i (s) X i,i (s)) dtds 1/ 1/ X0 (t)(x i (s) X 1/ i,i(s)) dtds

I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 389 = ( X 0 (t))1/ ((X i (s) X i,i (s)) ) 1/ dtds X0 (t)dt = X 0 X 0 X 0,i < (X i (s) X i,i (s)) ds (.9) on account of (1.4). This completes the proof of (.3). Since X 0,m (t)x 0,m (s) = X 0 (t)x 0 (s), in order to establish (.4), it is enough to show that m [X 0,m (t)x i,m (s)] dtds <. It follows from the definition of X i,m that the vectors (X 0,m, X i,m ) and (X 0, X i,m ) have the same distribution for all 1 i m. Also, (X i,m, X i,i ) has the same distribution as (X 0, X 0,i ), 1 i m. Hence following the arguments in (.9) we get 1/ m X 0,m (t)x i,m (s) dtds m = X 0 (t)x i,m (s) dtds X 0 m (X i,m (s) X i,i (s)) ds X 0 X 0 X 0,i <. The proof of (.4) is now complete. The arguments used above also prove (.5). Repeating the previous arguments we have C(t, t)dt = = 1/ X0 (t)dt + [X 0 (t)x i (t)] dt X0 (t)dt + [X 0 (t)(x i (t) X i,i (t))] dt X0 (t)dt + ( X0 (t))1/ ([X i (t) X i,i (t)] ) 1/ dt X 0 + X 0 (t)dt 1/ 1/ [X i (t) X i,i (t)] dt

390 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 Observing that C(t, t)dt = = X 0 + ( X 0 ) 1/ ( X 0 X 0,i ) 1/ <. λ i φi (t)dt = λ i, the proof of (.6) is complete. The same arguments can be used to establish (.7). The relation in (.8) can be established along the lines of the proof of (.5). By the Karhunen Loéve expansion, we have that Define X i,m (t) = X i,m,k (t) = X i,m, φ l,m φ l,m (t). (.10) K X i,m, φ l,m φ l,m (t) (.11) to be the partial sums of the series in (.10), and X i,m,k (t) = X i,m (t) X i,m,k (t) = X i,m, φ l,m φ l,m (t). (.1) Lemma.3. If {Z i } are independent L valued random variables such that Z 1 (t) = 0 and Z 1 <, (.13) then for all x > 0 we have that P max k Z 1 k i > x 1 x Z i. (.14) Proof. Let F k be the sigma algebra generated by the random variables {Z j } k j=1. By assumption (.13) and the independence of the Z i s we have that k+1 Z i F k k = Z i + Z k+1 k Z i. Therefore k is a non-negative submartingale with respect to the filtration Z i k=1 {F k } k=1. If we define A = ω : max k Z 1 k i > x,

I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 391 then it follows from Doob s maximal inequality [6, p. 47] that x P max k Z 1 k i > x Z i I A which completes the proof. Z i, Lemma.4. If (1.1) (1.4) hold, then for all x > 0, 1 k lim lim sup P max X K 1 k i,m,k > x = 0. (.15) Proof. Define Q k ( j) = {i : 1 i k, i = j (mod m)} for j = 0, 1,..., m 1, and all positive integers k. It is then clear that k X i,m,k = m 1 j=0 i Q k ( j) X i,m,k. We thus obtain by the triangle inequality that 1 k P max X 1 k i,m,k > x P m 1 j=0 max 1 k 1 It is therefore sufficient to show that for each fixed j, 1 lim lim sup P max X K 1 k i,m,k > x = 0. i Q k ( j) i Q k ( j) X i,m,k > x By the definition of Q k ( j), { X i,m,k } i Qk ( j) is an iid sequence of random variables. So, by applications of Lemma.3 and assumption (1.3), we have that P max 1 X 1 k i,m,k > x 1 x 1 X i,m,k i Q k ( j) 1 x X 0,m,K = 1 x i Q ( j) λ l,m. (.16) Since the right hand side of (.16) tends to zero as K tends to infinity independently of, (.15) follows. Clearly, with k = x we have 1 k K 1 x X i,m,k (t) = X i,m, φ j,m φ j,m (t). (.17) j=1.

39 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 Lemma.5. If (1.1) (1.4) hold, then the K dimensional random process 1 x 1 x X i,m, φ 1,m,..., X i,m, φ K,m converges, as, in D K [0, 1] to λ 1/ 1,m W 1(x),..., λ 1/ K,m W K (x), (.18) where {W i } K are independent, identically distributed Wiener processes. 1 x Proof. A similar procedure as in Lemma.4 shows that for each j, X i,m, φ j,m can be written as a sum of sums of independent and identically distributed random variables, and thus, by Billingsley [3], it is tight. This implies that the K dimensional process 1 x 1 x X i,m, φ 1,m,..., X i,m, φ K,m is tight, since it is tight in each coordinate. Furthermore, the Cramér Wold device and the central limit theorem for m-dependent random variables (cf. [7, p. 119]) shows that the finite dimensional distributions of the vector process converge to the finite dimensional distributions of the process in (.18). The lemma follows. In light of the Skorokhod Dudley Wichura theorem (cf. [6, p. 47]), we may reformulate Lemma.5 as follows. Corollary.1. If (1.1) (1.4) hold, then for each positive integer, there exists K independent, identically distributed Wiener processes {W i, } K such that for each j, 1 x sup X i,m, φ j,m λ 1/ j,m W P j, (x) 0, 0 x 1 as. Lemma.6. If (1.1) (1.4) hold, then for {W i, } K defined in Corollary.1, we have that 1 x sup 0 x 1 as. X i,m,k (t) Proof. By using (.17), we get that K 1 x K X i,m,k (t) λ 1/ l,m W l, (x)φ l,m (t) = K 1 x X i,m, φ l,m λ 1/ l,m W l, (x) φ l,m (t). λ 1/ l,m W P l, (x)φ l,m (t) dt 0, (.19)

I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 393 The substitution of this into the expression in (.19) along with a simple calculation shows that 1 x sup X i,m,k (t) 0 x 1 K 1 x = sup 0 x 1 K sup 0 x 1 as, by Corollary.1. K λ 1/ l,m W l, (x)φ l,m (t) dt X i,m, φ l,m λ 1/ l,m W l, (x) 1 x X i,m, φ l,m λ 1/ l,m W l, (x) Lemma.7. If (1.1) (1.4) hold, sup 0 x 1 λ 1/ l,m W l(x)φ l,m (t) dt P 0, P 0, (.0) as K, where W 1, W,... are independent and identically distributed Wiener processes. Proof. Since the functions {φ l,m } are orthonormal, we have that sup λ 1/ l,m W l(x)φ l,m (t) dt = sup 0 x 1 0 x 1 as K. Therefore (.0) follows from Markov s inequality. λ l,m W l (x) λ l,m sup Wl (x) 0, 0 x 1 Lemma.8. If (1.1) (1.4) hold, then for each we can define independent identically distributed Wiener processes {W i, } K such that 1 x sup 0 x 1 as. X i,m (t) Proof. It follows from Lemmas.4.7. λ 1/ l,m W P l, (x)φ l,m (t) dt 0, Since the distribution of W l,, 1 l < does not depend on, it is enough to consider the asymptotics for λ 1/ l,m W l(x)φ l,m (t), where W l are independent Wiener processes. Lemma.9. If (1.1) (1.4) hold, then for each m we can define independent and identically distributed Wiener processes W l,m (x), 1 l < such that sup 0 x 1 as m. λ 1/ l,m W l(x)φ l,m (t) λ 1/ l W l,m (x)φ l (t) dt P 0, (.1)

394 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 Proof. Let m (x, t) = λ 1/ l,m W l(x)φ l,m (t). Let M be a positive integer and define x i = i/m, 0 i M. It is easy to see that max sup ( m (x i + h, t) m (x i, t)) dt 0 i<m 0 h 1/M = λ l,m max max 0 i<m sup 0 i<m 0 h 1/M (W l (x i + h) W l (x i )) 0 h 1/M(W sup 1 (x i + h) W 1 (x i )) λ l,m. Using Lemma. we get that λ l,m = m (1, t)dt = C m (t, t)dt C(t, t)dt = λ l. So by the modulus of continuity of the Wiener process (cf. [15]) we get that lim lim sup max sup ( m (x i + h, t) m (x i, t)) dt = 0. (.) M m 0 i<m 0 h 1/M By the Karhunen Loéve expansion we can also write m as m (x, t) = m (x, ), φ l φ l (t) and m (x, t)dt = ( m (x, ), φ l ). So by Lemma. we have ( m (x, ), φ l ) x λ l. Also, for any positive integer l, ( m (x, ), φ l ) = C m (t, s)φ l (t)φ l (s)dtds C(t, s)φ l (t)φ l (s)dtds = λ l, as m. Hence for every z > 0 we have lim sup lim sup P m (x, ), φ l φ l (t) dt > z = 0. (.3) K m The joint distribution of (x i, ), φ l, 1 i M, 1 l K is multivariate normal with zero mean. Hence they converge jointly to a multivariate normal distribution. To show their joint

I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 395 convergence in distribution, we need to show the convergence of the covariance matrix. Using again Lemma. we get that (x i, ), φ l (x j, ), φ k = min(x i, x j ) C m (t, s)φ l (t)φ k (s)dtds min(x i, x j ) C(t, s)φ l (t)φ k (s)dtds = min(x i, x j )λ l I {k = l}. Due to this covariance structure and the Skorokhod Dudley Wichura theorem (cf. [6, p. 47]) we can find independent Wiener processes W l,m (x), 1 l < such that max 1 i M max (x i, ), φ l λ 1/ l W l,m (x i ) = o P (1), as m. 1 l K Clearly, for all 0 x 1 W l,m (x)φ l (t) dt = x λ 1/ l and therefore similarly to (.3) lim sup lim sup P K m λ 1/ l for all z > 0. Similarly to (.) one can show that max sup 0 i<m 0 h 1/M max 0 i<m λ l 0, as m, W l,m (x)φ l (t) dt > z = 0 ( W l,m (x i + h) W l,m (x i ))φ l (t) dt 0 h 1/M(W sup (x i + h) W (x i )) λ l 0, as M, where W is a Wiener process. This also completes the proof of Lemma.9. Proof of Theorem 1.1. First we approximate S (x,t) with m-dependent processes (Lemma.1). The second step of the proof is the approximation of the sums of m-dependent processes with a Gaussian process with covariance function min(x, x )C m (t, s), where C m is defined in (.) (Lemma.8). The last step of the proof is the convergence of Gaussian processes with covariance functions min(x, x )C m (t, s) to a Gaussian process with covariance function min(x, x )C(t, s) (Lemma.9). 3. Some moment and maximal inequalities In this section we give the proof of the maximal inequality used in Lemma.1 which is a crucial ingredient of the proof of Theorem 1.1. Actually, we will prove below some moment and maximal inequalities for partial sums of function valued Bernoulli shift sequences which have their own interest and can be used in various related problems. Our first lemma is a Hilbert space version of Doob s [1, p. 6] inequality.

396 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 Lemma 3.1. If Z 1 and Z are independent mean zero Hilbert space valued random variables, and if 0 < δ 1, then Z 1 + Z +δ Z 1 +δ + Z +δ + Z 1 ( Z ) δ/ + Z ( Z 1 ) δ/. Proof. Since 0 < δ 1, for any A, B 0 we have that (A + B) δ A δ + B δ (cf. [17, p. 3]). An application of this inequality along with Minkowski s inequality gives that Z 1 + Z δ ( Z 1 + Z ) δ Z 1 δ + Z δ. We also have by Hölder s inequality that Z 1 δ ( Z 1 ) δ/. This yields that Z 1 + Z +δ = Z 1 + Z Z 1 + Z δ which proves the lemma. Z 1 + Z ( Z 1 δ + Z δ ) = [ Z 1 + Z + Z 1, Z ]( Z 1 δ + Z δ ) = Z 1 +δ + Z +δ + Z 1 Z δ + Z Z 1 δ Z 1 +δ + Z +δ + Z 1 ( Z ) δ/ + Z ( Z 1 ) δ/, Remark 3.1. If Z 1 and Z are independent and identically distributed, then the result of Lemma 3.1 can be written as Let Z 1 + Z +δ Z 1 +δ + ( Z 1 ) 1+δ/. I (r) = ( X 0 X 0,l r ) 1/r. (3.1) We note that by (1.4), I (r) < for all r + δ. Lemma 3.. If (1.1) (1.4) hold, then we have n (X i X i,m ) n A, where A = (X 0 (t) X 0,m (t)) dt + 5/ (X 0 (t) X 0,m (t)) dt 1/ I (). (3.) Proof. Let i = X i X i,m. By Fubini s theorem and the fact that the random variables are identically distributed, we conclude

I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 397 n n i = i (t) dt n 1 = n 0 (t)dt + (n i)[ 0 (t) i (t)]dt n n 0 n 1 (t)dt + n [ 0 (t) i (t)] dt 0 (t)dt + n [ 0 (t) i (t)] dt. (3.3) We recall X i,i from (1.4). Under this definition, the random variables 0 and X i,i are independent for all i 1. Let Z i = X i,m, if i > m and Z i = g(ϵ i,..., ϵ 1, δ i ), if 1 i m, where δ i = (δ i,0, δ i, 1,...) and δ i, j are iid copies of ϵ 0, independent of the ϵ l s and ϵ k,l s. Clearly, Z i and 0 are independent and thus with i,i = X i,i Z i we have [ 0 (t) i (t)] = [ 0 (t)( i (t) i,i (t))]. Furthermore, by first applying the Cauchy Schwarz inequality for expected values and then by the Cauchy Schwarz inequality for functions in L, we get that 1/ [ 0 (t)( i (t) i,i (t))] dt 0 (t) i (t) i,i (t) 1/ dt 1/ 0 (t)dt i (t) i,i (t) 1/ dt. Also, i (t) i,i (t) dt X i (t) X i,i (t) dt + X i,m (t) Z i (t) dt. The substitution of this expression into (3.3) gives that n 1/ i n 0 (t)dt + 3/ n 0 (t)dt X i (t) X 1/ i,i(t) dt + n X i,m (t) Z i (t) dt 1/ 0 (t)dt + 5/ 0 (t)dt 1/ I (), which completes the proof.

398 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 Theorem 3.1. If (1.1) (1.4) hold, then for all 1 +δ (X i X i,m ) 1+δ/ B, where B = X 0 X 0,m +δ + c +δ δ [A 1+δ/ + J +δ m + J m A (1+δ)/ + A (1+δ/)δ J m ] + (c δ J m )1/(1 δ) with A defined in (3.), c δ = 36 1 1 1 δ/ (3.5) and J m = ( X 0 X 0,m +δ ) (κ δ)/(κ(+δ)) ( X 0 X 0,l +δ ) 1/κ. Proof. We prove Theorem 3.1 using mathematical induction. By the definition of B, the inequality is obvious when = 1. Assume that it holds for all k which are less than or equal to 1. We assume that is even, i.e. = n. The case when is odd can be done in the same way with minor modifications. Let i = X i X i,m. For all i satisfying n + 1 i n, we define X i,n = g(ϵ i, ϵ i 1,..., ϵ n+1, ϵ n, ϵ n 1,...) where the ϵ j s denote iid copies of ϵ 0, independent of {ϵ i, < i < } and {ϵ k,l, < k, l < }. We define Z i,n = X i,m, if m + n + 1 i n and Z i,n = g(ϵ i,..., ϵ n+1, ϵ n,... ϵ i m+1, δ i) with δ i = (δ i,n, δ i,n 1,...), if n + 1 i n + m, where the δ k,l s are iid copies of ϵ 0, independent of the ϵ k s and ϵ k,l s. Let i,n = X i,n Z i,n, if n + 1 i n. Under this definition, the sequences { i, 1 i n} and {i,n, n + 1 i n} are independent and have the same distribution. Let n n Θ = i + i,n n i,n and Ψ = i. i=n+1 i=n+1 By applying the triangle inequality for L and expected values, we get n +δ n n n i = i + i,n + i i,n +δ i=n+1 i=n+1 (3.4) (Θ + Ψ) +δ (Θ +δ ) 1/(+δ) + (Ψ +δ ) 1/(+δ) +δ. (3.6) The two-term Taylor expansion gives for all a, b 0 and r > that (a + b) r a r + ra r 1 b + r(r 1) (a + b) r b. (3.7)

I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 399 Since both of the expected values in the last line of the inequality in (3.6) are positive, we obtain by (3.7) that n +δ i Θ +δ + ( + δ)(θ +δ ) (1+δ)/(+δ) (Ψ +δ ) 1/(+δ) + ( + δ)(1 + δ) (Θ +δ ) 1/(+δ) + (Ψ +δ ) 1/(+δ) δ (Ψ +δ ) /(+δ). (3.8) We proceed by bounding the terms (Ψ +δ ) 1/(+δ), and Θ +δ individually. Applications of both the triangle inequality for L and for expected values yield that (Ψ +δ ) 1/(+δ) n = i +δ 1/(+δ) i,n n i=n+1 i=n+1 +δ i i,n i=n+1 n ( i i,n +δ ) 1/(+δ). By Hölder s inequality we have, with κ in (1.4), 1/(+δ) ( i i,n +δ ) 1/(+δ) = ([ i i,n (+δ) /κ i i,n (+δ) (+δ) /κ ]) 1/(+δ) ( i i,n +δ ) 1/κ ( i i,n +δ ) (κ δ)/(κ(+δ)). It follows from the definition of i, i,n and the convexity of x+δ that and Thus we get i i,n +δ 1+δ ( X i X i,n +δ + X i,m Z i,n +δ ) +δ X 0 X 0,i n +δ i i,n +δ 1+δ ( X i X i,m +δ + X i,n Z i,n +δ ) +δ X 0 X 0,m +δ. (Ψ +δ ) 1/(+δ) ( X 0 X 0,m +δ ) (κ δ)/(κ(+δ)) ( X 0 X 0,l +δ ) 1/κ = J m. To bound Θ +δ, since n i and n i=n+1 i,n are independent and have the same distribution, we have by Lemma 3., Remark 3.1 and the inductive assumption that Θ +δ n n +δ = i + i,n i=n+1

400 I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 n +δ n i + i n 1+δ/ B + (n A) 1+δ/. The substitution of these two bounds into (3.8) give that n +δ i n 1+δ/ B + (n A) 1+δ/ 1+δ/ + ( + δ)[n 1+δ/ B + (n A) 1+δ/ ] (1+δ)/(+δ) J m δ + ( + δ)(1 + δ) n 1+δ/ B + (n A) 1+δ/ + J m J m. (3.9) Furthermore, by the definition of B, we may further bound each summand on the right hand side of (3.9). We obtain for the first two terms that n 1+δ/ B + (n A) 1+δ/ (n) 1+δ/ B δ/ + A1+δ/ B (n) 1+δ/ B δ/ + 6cδ 1. A similar factoring procedure applied to the expression in the second line of (3.9) yields that ( + δ) n 1+δ/ B + (n A) 1+δ/ (1+δ)/(+δ) Jm 6 (n 1+δ/ B) (1+δ)/(+δ) + (n A) (1+δ/)[(1+δ)/(+δ)] J m (n) 1+δ/ 6J m B B 1/(+δ) + 6J m A (1+δ/)[(1+δ)/(+δ)] B (n) 1+δ/ B. 1c 1 δ Since 0 < δ < 1, the expression in the third line of (3.9) may be broken into three separate terms: δ ( + δ)(1 + δ) n 1+δ/ B + (n A) 1+δ/ + J m J m 6(n 1+δ/ B) δ J m + 6((n A)(1+δ/) ) δ J m + 6J +δ m. Furthermore by again applying the definition of B we have that 6(n 6(n 1+δ/ B) δ Jm 1+δ/ = B) δ J (n)1+δ/ m B (n) 1+δ/ B 6J (n) 1+δ/ B m B 1 δ (n) 1+δ/ B[6c 1 δ ], 6((n A) (1+δ/) ) δ J m = (n)1+δ/ B 6((n A) (1+δ/) ) δ J m (n) 1+δ/ B

and 6J +δ m I. Berkes et al. / Stochastic Processes and their Applications 13 (013) 385 403 401 (n) 1+δ/ 6A (1+δ/)δ Jm B B (n) 1+δ/ B[6c 1 δ ], = (n) 1+δ/ 6J +δ m B (n) 1+δ/ B (n) 1+δ/ B The application of these bounds to the right hand side of (3.9) give that n +δ i (n) 1+δ/ B δ/ + 36cδ 1 = (n) 1+δ/ B, which concludes the induction step and thus the proof. 6J +δ m (n) 1+δ/ B[6cδ 1 ]. B Theorem 3.. If (1.1) (1.4) hold, then we have k +δ (X i X i,m ) a m 1+δ/ (3.10) max 1 k with some sequence a m satisfying a m 0 as m. Proof. By examining the proofs, it is evident that Theorem 3.1 in [4] holds for L valued random variables. Furthermore, by the stationarity of the sequence {X i X i,m } and Theorem 3.1, the conditions of Theorem 3.1 in Móricz are satisfied and therefore k +δ (X i X i,m ) c δ 1+δ/ B, max 1 k with some constant c δ, depending only on δ and B is defined in (3.4). Observing that B = B m 0 as m, the result is proven. Theorem 3.1 provides inequality for the moments of the norm of partial sums of X i X i,m which are not Bernoulli shifts. However, checking the proof of Theorem 3.1, we get the following result for Bernoulli shifts. Theorem 3.3. If (1.1), (1.3) are satisfied and X is a Bernoulli shift satisfying I ( + δ) = ( X 0 X 0,l +δ ) 1/(+δ) < with some 0 < δ < 1, where X 0,l is defined by (1.4), then for all 1 +δ X i 1+δ/ B, where B = X 0 +δ + cδ +δ [A 1+δ/ + I ( + δ)a (1+δ)/ + I +δ ( + δ) + A (1+δ/)δ I ( + δ)] + (c δ I ()) 1/(1 δ),

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