A Berry-Esse\'en theorem for partial sums of functionals of heavy-tailed moving averages

In this paper we obtain Berry-Esse\'en bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $\alpha$, and its tail-index, which is controlled by a parameter $\beta$. In fact, we obtain the classical $\sqrt{1/ n}$ rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when $\alpha\beta>3$ or $\alpha\beta>4$ in the case of Wasserstein and Kolmogorov distance, respectively. Our quantitative bounds rely on a new second-order Poincare inequality on the Poisson space, which we derive through a combination of Stein's method and Malliavin calculus. This inequality improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory Relat. Fields 165 (2016)].


Overview
The main goal of this paper is to characterize the convergence rates associated with asymptotic normality of a class of statistics of Lévy moving averages. For processes with finite fourth moments, Theorem 8.2 in [17] obtains rates for a class of specific examples. Its proof relies on second-order Poincaré inequalities on the Poisson space [17, Theorem 1.1-1.2], which in turn are based on the celebrated Malliavin-Stein method.
The main novelty of our results and methodology is the study of convergence rates for processes having heavy tails and strong memory, as e.g. the linear fractional stable noise or fractional stable ARIMA processes. In fact, in our setting the upper bounds in the second-order Poincaré inequalities obtained in [17] may converge to infinity after the application of our standard estimate (4.4) to them. As a consequence, we develop a new modified second-order Poincaré inequality on the Poisson space, which allows us to efficiently bound Wasserstein and Kolmogorov distances associated with normal approximation of a class of statistics of Lévy moving averages. The improved bounds are important in their own right as they may prove to be useful in other contexts, where the considered stochastic process exhibits heavy tails and strong memory.

Background
The Berry-Esseén theorem gives a quantitative bound for the convergence rate in the classical central limit theorem. To be more precise, let (X i ) i∈N be a sequence of independent and identically distributed (i.i.d.) random variables with mean zero, variance one and finite third moment, and set V n = n −1/2 n i=1 X i and Z ∼ N (0, 1). Then, the central limit theorem says that V n d −→ Z as n → ∞, where d −→ denotes convergence in distribution. A natural next question is to ask for quantitative bounds between V n and Z, that is, how far is V n from Z in a certain sense. An answer to this question is provided by the Berry-Esseén theorem, which states that where d K denotes the Kolmogorov metric between two random variables and where C is a constant depending on the third moment of the underlying random variables. The Berry-Esseén bound (1.1) is optimal in the sense that there exist random variables as above such that d K (V n , Z) is bounded from below by a constant times n −1/2 , see e.g. [11, (5.26.2)].
The situation when the summands (X i ) i∈N are dependent is much more complicated, compared to the classical i.i.d. setting. One of the most important models in this situation is the fractional Gaussian noise, which we will describe in the following. For H ∈ (0, 1), the fractional Brownian motion is the unique centered Gaussian process (Y t ) t∈R with covariance function cov(Y t , Y u ) = 1 2 |t| 2H + |u| 2H − |t − u| 2H , for all t, u ∈ R.
The fractional Gaussian noise (X n ) n∈Z is the corresponding increment process X n = Y n − Y n−1 . Let

Heavy-tailed moving averages
Let us now describe our results in more detail. We consider a two-sided Lévy process L = (L t ) t∈R with no Gaussian component, L 0 = 0 a.s. and Lévy measure ν, that is, for all θ ∈ R, the characteristic function of L 1 is given by where b ∈ R, and χ is a truncation function, i.e. a bounded measurable function such that χ(x) = 1 + o(|x|) as x → 0 and χ(x) = O(|x| −1 ) as |x| → ∞. We assume that the Lévy measure ν has a density κ satisfying κ(x) ≤ C|x| −1−β for all x ∈ R \ {0}, (1.4) for β ∈ (0, 2) and a constant C > 0. We consider a Lévy moving average of the form (1.5) where g : R → R is a measurable function such that the integral exists, see [35] for sufficient conditions. Lévy moving averages are stationary infinitely divisible processes, and are often used to model long-range dependence and heavy tails. When the Lévy process L is symmetric, i.e. when −L 1 equals L 1 in distribution, a sufficient condition for X to be well-defined is that R |g(s)| β ds < ∞, due to assumption (1.4). Throughout the paper we will assume that the kernel function g satisfies for all x > 0, (1.6) for some finite constants K > 0, α > 0 and γ ∈ R. We refer to Subsections 1.
based on a measurable function f : R → R with E[|f (X 1 )|] < ∞. Typical examples, which are important in statistics, are the empirical characteristic functions (f : x → e iθx , where θ ∈ R), the empirical distribution functions (f : x → 1 (−∞,t] (x), where t ∈ R), and power variations (f : x → |x| p , where p > 0). For example, in a recent paper [22] the empirical characteristic function has been successfully employed to estimate the parameters of a linear fractional stable motion observed at high or low frequency.
The major breakthrough on establishing central limit theorems for V n was achieved in the paper Hsing [12,Theorem 1], and was extended in [32,33,22,2,1], whereas non-central limit theorems for V n are established in [1,2,41,42]. From these results, it follows that if (X t ) is given by (1.5) with L being a β-stable Lévy process and the kernel function g satisfying (1.6) with γ ≥ 0 and αβ > 2 we have that

Main results
To present our main result let C 2 b (R) denote the space of twice continuously differentiable functions such that f , f and f are bounded. Our result reads as follows: Theorem 1.1. Let (X t ) t∈R be a Lévy moving average given by (1.5), satisfying (1.4) for some 0 < β < 2, and (1.6) with αβ > 2 and γ > −1/β. Let V n be the corresponding partial sums of functionals, given by (1.7), based on f ∈ C 2 b (R). Also, let Z ∼ N (0, 1) be a standard Gaussian random variable, and set v n = var(V n ) for all n ∈ N. (1.9) and the series (1.9) converges absolutely. Suppose that v > 0. Then, v −1 n V n d −→ N (0, 1) as n → ∞. Moreover, for each n ∈ N, where C > 0 is a constant that does not depend on n and d W denotes the Wasserstein distance.

Remark 1.2.
In the following we will make a few remarks on Theorem 1.1.
1. The bounds on the Wasserstein and Kolmogorov distances to the normal distribution, depend on the interplay between memory of X, which is controlled by α, and the tail-index of X, which is controlled by β. In fact, we obtain the classical 1/ √ n rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when αβ > 3 or αβ > 4 in the case of Wasserstein and Kolmogorov distance, respectively. We conjecture that our bounds are optimal in this case. We note also that all rates in Theorem 1.1 converge to zero. For 2 < αβ < 3, the bound on the Kolmogorov distance (1.11) follows from the bound (1.10) on the Wasserstein distance via the inequality 4. Our proof of Theorem 1.1 relies heavily on the smoothness of f ∈ C 2 b (R) through the estimates (4.4)-(4.5). However, since the CLT (1.8) holds whenever f is bounded and measurable, it would be interesting to obtain quantitative bounds for less regular f . Moreover, within the framework of Gaussian processes, there has been recent advances in multivariate quantitative bounds and functional central limit theorems, see [25,3], and it would be of interest to obtain such extensions for Poisson functionals also. In particular, we note that the CLT (1.8) holds in a multivariate setting, cf. [32, Theorem 2.1], but presently no multivariate quantitative bounds exists.
In the following we will apply Theorem 1.1 to the four important examples: linear fractional stable noises, fractional Lévy noises, stable fractional ARIMA processes, and stable Ornstein-Uhlenbeck processes. Throughout we will fix the notation used in , v 2 given in (1.9) satisfies v 2 > 0, and Z ∼ N (0, 1) is a standard Gaussian random variable.

Linear fractional stable noises
Our first example concerns the linear fractional stable noise. To define this process we let L be a β-stable Lévy process with β ∈ (0, 2), and (1.13) where H ∈ (0, 1). For β = 1 we assume furthermore that L is symmetric, that is, L 1 equals −L 1 in distribution. The linear fractional stable motion (Y t ) t∈R has stationary increments and is self-similar with index H, and can be viewed as a heavy-tailed extension of the fractional Brownian motion, see [39] for more details. In this setting we deduce that α = 1 − H + 1/β and the condition αβ > 2 translates to β ∈ (1, 2), 0 < H < 1 − 1/β. Since β > 1 we never have αβ ≥ 3. Corollary 1.3. Let (X t ) t∈R be the linear fractional stable noise defined as in (1.13). For β ∈ (1, 2) and 0 < H < 1 − 1/β we have that where C > 0 is a constant not depending on n.

Stable fractional ARIMA processes
In the following we will consider the stable fractional ARIMA process. To this end, we let p, q ∈ N, and Φ p and Θ q be polynomials with real coefficients on the form where we assume that Φ p and Θ q do not have common roots, and that Φ p has no roots in the closed unit disk {z ∈ C : |z| ≤ 1}. The stable fractional ARIMA(p, d, q) process (X n ) n∈N is the solution to the equation where ( n ) n∈N are independent and identically symmetric β-stable random variables with β ∈ (0, 2), B denotes the backshift operator, and d ∈ R \ Z. The equation should be understood as in [13,Section 2]. For d < 1 − 1/β, there exists a unique solution (X n ) n∈N to (1.15), and it is a discrete moving average of the form where c 0 denotes a positive constant, cf. Theorem 2.1 of [13]. Notice that the process (X n ) n∈N can be written in distribution as ) (x) and L being a symmetric β-stable Lévy process. Here α = 1 − d and by Theorem 1.1 we obtain the following result.
where C > 0 is a constant not depending on n.

Stable Ornstein-Uhlenbeck processes
In our last example we will consider a stable Ornstein-Uhlenbeck process (X t ) t∈R , given where L denotes a β-stable Lévy process with β ∈ (0, 2), and λ > 0 is a finite constant. In this case α > 0 can be chosen arbitrarily large and we obtain the following result.
Corollary 1.6. Let (X t ) t∈R be a stable Ornstein-Uhlenbeck process given by (1.16).

Structure of the paper
The paper is structured as follows. Section 2 presents a related result and some discussions. Basic notions of Malliavin calculus on Poisson spaces and the new bounds for the Wasserstein and Kolmogorov distances are demonstrated in Section 3. In Section 4 we prove Theorem 1.1 based on the general bounds obtained in Section 3.

Related literature and discussion
Normal approximation of non-linear functionals of Poisson processes defined on general state spaces has become a topic of increasing interest during the last years. In particular, quantitative bounds for normal approximations were obtained by combining Malliavin calculus on the Poisson space with Stein's method. The resulting bounds have successfully been applied in various contexts such as stochastic geometry (see, e.g., [8,14,15,17,20,38]), the theory of U-statistics (see, e.g., [8,9,10,38]), non-parametric Bayesian survival analysis (see [28,29]) or statistics of spherical point fields (see, e.g., [4,5]). We refer the reader also to [26], which contains a representative collection of survey articles.
The first quantitative bounds for asymptotically normal functionals of Lévy moving averages have been derived in [17]. We briefly introduce their framework, but phrase their results in an equivalent way through Lévy processes instead of Poisson random measures. Let (X t ) t∈R denote a Lévy moving average of the form (1.5) where L is centered. Assume that the Lévy measure ν satisfies the condition which implies that (L t ) t∈R is of locally bounded variation with a finite second moment. Suppose that the kernel function g satisfies the condition R |g(x)| + g(x) 2 dx < ∞. The functional under consideration is defined by which can be interpreted as the continuous version of the statistic V n . We now state Theorem 8.2 of [17] in the case of p = 0.
Let Z be a standard Gaussian random variable. Then there exists a constant C > 0 such that for all T ≥ t 0 , While Theorem 2.1 (and its proof) relies heavily on a finite fourth moment, Theorem 1.1 works for infinite variance models, as e.g. stable processes. The heavy tails of these processes force us to introduce the improved version of the bounds in [17, Theorems 1.1 and 1.2] in the next section, and they are also responsible for slower rates of convergence in Theorem 1.1 compared to Theorem 2.1.

New bounds for normal approximation on Poisson spaces
The aim of this section is to introduce new bounds on the Wasserstein and the Kolmogorov distances between a Poisson functional and a standard Gaussian random variable. Although similar bounds were previously derived in [17,27], they are not sufficient in certain settings. For this reason, we shall provide an improved version, which is adapted to our needs. Since such a bound might be useful in other contexts as well, we formulate and prove it in a general set-up, which is specialised later in this paper.

Poisson spaces and Malliavin calculus
We recall some basic notions of Malliavin calculus on Poisson spaces and refer the reader to [18,26] for further background information. We fix an underlying probability space (Ω, A, P), let (X, X ) be a measurable space and λ be a σ-finite measure on X (in the applications we consider here X will be of the form R × R). By η we denote a Poisson process on X with intensity measure λ, see [18] for a formal definition and an explicit construction of such a process. We often consider η as random element in the space of integer-valued σ-finite measures on X, denoted by N, which is equipped with the σ-algebra generated by all evaluations µ → µ(A) for A ∈ X . A real-valued random variable F is called a Poisson functional if there exists a measurable function φ : N → R such that P-almost surely F = φ(η). We let L 2 η denote the space of all square-integrable Poisson functionals F = φ(η). It is well known that each F ∈ L 2 η admits a chaotic decomposition I m (f m ), (3.1) where I m denotes the mth order integral with respect to the compensated (signed) measureη = η − λ and f m : X m → R are symmetric functions with f m ∈ L 2 (λ m ).
Here, for a measure µ on X and k ∈ N we write L 2 (µ k ) for the space of functions f : X k → R, which are square integrable with respect to the k-fold product measure of µ, and · L 2 (µ k ) for the corresponding L 2 -norm. For z ∈ X and a Poisson functional the Malliavin derivative of F in direction z, also known as the difference operator, or in a geometric setting, the add-one cost operator. Here, δ z stands for the Dirac measure at z ∈ X. We can consider DF as a function on Ω × X acting as (ω, z) → D z F (ω). If DF is square integrable with respect to the product measure P ⊗ λ we shall write DF ∈ L 2 (P ⊗ λ) in what follows. Finally, let us define the second-order Malliavin derivative of F with respect to two points z 1 , z 2 ∈ X by putting (note that this definition is symmetric in z 1 and z 2 ). The Kabanov-Skorohod integral δ maps random functions u from L 2 η (P ⊗ λ) to random variables in L 2 η . To introduce the definition of the operator δ, let The domain dom δ of δ consists of all random functions u that satisfy the condition whereh m denotes the symmetrisation of the function h m . For u ∈ dom δ the Kabanov-Skorohod integral is defined by Finally, we introduce the Ornstein-Uhlenbeck generator L and its (pseudo) inverse L −1 .
The domain dom L of L consists of all elements F ∈ L 2 η with chaotic decomposition (3.1) that additionally satisfy the condition EJP 25 (2020), paper 31.
The important relationships between the introduced operators can be summarised as follows: where u ∈ dom δ. We refer to [26] or [27] for a more detailed exposition.

Wasserstein distance
In this subsection we derive quantitative bounds for the Wasserstein distance. We recall that the Wasserstein distance between two random variables F and G is defined where the supremum is running over all Lipschitz functions h : R → R with a Lipschitz constant less than or equal to 1. To formulate the next result we introduce the three quantities (although γ 1 , γ 2 and γ 3 depend on the Poisson functional F , we suppress this dependency in our notation for simplicity). The theorem below is an improved version of the secondorder Poincaré inequality for Poisson functionals from [17, Theorem 1.1], where the main difference stems from the term γ 3 .
. It turns out that our improvement is absolutely crucial as the quantity introduced in [17, Theorem 1.1] converges to infinity in our setting.
In our framework the main problem appears when the term |D z V n | becomes large, say, larger than 1. Such an event has a relatively high weight under the measure λ. Apart from the application of Malliavin calculus another ingredient on which the proof of Theorem 3.1 is build is Stein's method for normal approximation (we refer to EJP 25 (2020), paper 31.
Starting with this estimate, we can now present the proof of Theorem 3.1.
Proof of Theorem 3.1. Let f ∈ F W and fix a, b ∈ R. Then using the bound for f we observe that Similarly, using the bound for f we have, by Taylor approximation, Next, using the definition of the Malliavin derivative and a Taylor expansion of f around F we see that where in view of the above considerations the remainder term R(·) satisfies the estimate |R(y)| ≤ min(2|y|, y 2 ) for all y ∈ R. Applying this together with the three relations for the Malliavin operators presented in the previous subsection, we see that EJP 25 (2020), paper 31.
As a consequence, where we used that f ∞ ≤ 1. Next, we apply [17, Proposition 4.1] to conclude that The proof is thus complete.

Kolmogorov distance
Now we turn our attention to the Kolmogorov distance between the two random variables F and Z, which is defined as Let f = f x be the bounded solution of the Stein's equation associated with the function 1 (−∞,x] for a given x ∈ R. It is well known that this solution satisfies the inequalities (we interpret f as the left-sided derivative at the point x, where f is not differentiable). Hence, with F K denoting the class of all absolutely continuous functions satisfying (3.7) we have N (0, 1). To obtain modified bounds for the Kolmogorov distance we introduce an arbitrary measurable function ϕ : R → R such that 0 ≤ ϕ ≤ 1. We may simply use ϕ = 1 [−1,1] or a smooth function with compact support. As discussed in Remark 3.2 we will use the function ϕ to distinguish between large and small values of the quantity |D z V n |.
To formulate the analogue of Theorem 3.1 for the Kolmogorov distance we introduce the quantities (again, we suppress in our notation the dependency on the Poisson functional F ).
We are now prepared to present our general estimate for the Kolmogorov distance between a Poisson functional and a standard Gaussian random variable.
where the decomposition is derived in [10, Equation (3.4)]. We have already seen in (3.6) that the inequality holds, where we have used that f ∞ ≤ 1. To treat the second term of the decomposition we need to distinguish whether D z F takes small or large values. In particular, we have EJP 25 (2020), paper 31.
where the second inequality follows from the identity 1 = (1 − φ(D z F )) + φ(D z F ), the triangle inequality and f ∞ ≤ 1. Now, repeating the methods from the proof of [10, Theorem 3.1] (see pages 7-9 therein) for the last term in (3.11), we conclude that In the next step, we need to apply the ideas from [17] to the new bound at (3.12). We obtain that Similarly, we have that 4 , EJP 25 (2020), paper 31.
Moreover, we note that Next, we treat the last term in (3.12). We set g(z) = ϕ(D z F )(D z F )|D z L −1 F | and observe the inequality is valid, where we recall that δ(g) stands for the Kabanov-Skorohod integral of g. Furthermore, we have the Kabanov-Skorohod isometric formula see [16,Theorem 5]. Applying the Cauchy-Schwarz inequality we deduce that For the last term A 2 we conclude, using the inequality |D y (GH)| ≤ |HD y G| + |GD y H| + |D y HD y G|, Combining (3.10) and (3.12), we conclude the assertion of Theorem 3.3.

Proof of Theorem 1.1
All positive constants, which do not depend on n, are denoted by C although they may change from occasion to occasion. Furthermore, we assume without loss of generality that K = 1 in condition (1.6). We extend the definition of the kernel g to the whole real line by setting g(x) = 0 for x ≤ 0. To apply Theorems 3.1 and 3.3 it will be useful for us to represent the process (X t ) t∈R , in (1.5), in terms of an integral with respect to a Poisson where the integral is defined as in [37, p. 3236], and The integrals in (4.1) and (4.2) exist since X t is well-defined, cf. [35,Theorem 2.7]. By (4.1) it follows that X t is a Poisson functional for all t ∈ R, i.e. there exists a measurable mapping φ = φ t : N → R such that X t = φ(η). Throughout this section we will repeatedly use that for any measurable positive function f : which follows by assumption (1.4) on ν. Here and below, C will denote a strictly positive and finite constant whose value might change from occasion to occasion.

Preliminary estimates
We let z = (x, s) ∈ R 2 , z j = (x j , s j ) ∈ R 2 for j ∈ {1, 2, 3}, and f ∈ C 2 b (R). By the mean-value theorem and (4.1), we have that Again, by applying the mean-value theorem and using that f, f and f are bounded we obtain the estimate Notice that A n (z 1 , z 2 ) ≤ C min(A n (z 1 ), A n (z 2 )).
Proof. From the substitution u = k −1 s we obtain Moreover, by the same procedure as in (4.8) and using succesive substitutions we obtain the bound (4.7).

Lemma 4.2.
The series v 2 defined in Theorem 1.1 converges absolutely, and v 2 n → v 2 as n → ∞.
Proof. In the following we will show that ∞ j=1 |cov(f (X j ), f (X 0 ))| < ∞. To prove (4.9) we use the covariance identity from Theorem 5.1 in [19] to get (4.10) where (G u ) u∈[0,1] are certain σ-algebras (which will not be important for us). As noticed in [19, Proof of Theorem 1.4] we may always assume that we are in the setting of [19,Theorem 1.5]. By Cauchy-Schwarz inequality and the contractive properties of conditional expectation it follows from (4.10) that where we have used (4.3) in the third inequality, the equality follows by the definition of ρ j , and the last inequality follows by Lemma 4.1. Since αβ > 2, (4.11) implies (4.9).
By (4.9), the series v 2 converges absolutely. Moreover, the stationarity of (X j ) j∈N where the convergence follows by Lebesgue's dominated convergence theorem together with (4.9).

Bounding the Wasserstein distance
Since v 2 n = var(V n ) → v 2 as n → ∞, cf. Lemma 4.2, and v > 0 by assumption, we note that v n is bounded away from zero. Let γ 1 , γ 2 , γ 3 be defined in (3.2), (3.3) and (3.4) with F = V n . Then, by Theorem 3.1 and using that v n is bounded away from 0, we have that Using the estimates (4.4) and (4.5) we will now compute bounds for the quantities γ 1 , γ 2 and γ 3 appearing in the bound for the Wasserstein distance in Theorem 3.1. Notice that the right hand sides in both estimates (4.4) and (4.5) are deterministic, so the expectations in the definitions of γ 1 , γ 2 , γ 3 can be omitted. We start with the term γ 1 .
we deduce the following inequality by (4.4) and (4.5): By the substitution w 2 i = x 2 i y i for i ∈ {1, 2, 3} we see that for any y 1 , y 2 , y 3 > 0 it holds Indeed, we have that EJP 25 (2020), paper 31. since β ∈ (0, 2). Therefore, we deduce the estimate where the first equality follows by substitution, the next inequality follows by the change , and the last inequality follows from the symmetry ρ −k = ρ k . By Lemma 4.1, we have that ∞ k=0 ρ k < ∞, and hence, (4.13) completes the proof of the estimate γ 1 ≤ Cn −1/2 .
Using a similar reasoning, we can also bound the term γ 2 . Proof. Recall that By the inequality (4.5) we immediately conclude that As in the proof of Lemma 4.3 a substitution shows that for any y 1 , y 2 , y 3 > 0, Therefore, we have the estimate (4.14) Now, the inequality |xy| ≤ x 2 + y 2 , valid for all x, y ∈ R, implies which by (4.14), shows that by the same arguments as in the proof of Lemma 4.3. Since k≥0 ρ k < ∞, cf. Lemma 4.1, the estimate γ 2 ≤ Cn −1/2 follows from (4.15).
The final term γ 3 in the bound for the Wasserstein distance is more subtle. It is this term, which decays slower than n −1/2 for certain parameter regimes.

Lemma 4.5.
There exists a constant C > 0 such that Proof. Recalling the inequality (4.4), we have that The following bound used in the proof of Lemma 4.5 is stated separately as a lemma, since we will also use it in the proof of upper bound for the Kolmogorov distance. Lemma 4.6. Let p ∈ [0, 2] and q > 2. There exists a finite constant C such that Proof of Lemma 4.6. To obtain the upper bound for the right hand side we need to decompose the integral into different parts according to whether |x| ∈ (0, 1), |x| ∈ [1, n α ] or |x| ∈ (n α , ∞). Using the symmetry in x this means that min (|A n (x, s)| p , |A n (x, s)| q ) λ(ds, dx) =: I 1 + I 2 + I 3 .
where we used g(u) = 0 for all u < 0 in the first inequality and assumption (1.6) on g in the second inequality. The third inequality follows from the fact that α > 1, which is implied by the assumptions α > 2/β and β < 2. For γ < 0 we have where the first equality follows by substitution. For γ ≥ 0, we have the simple estimate n 0 |f 1 (s, x)| q ds ≤ Cnx q . Similarly, we have that n 0 |f 2 (s, x)| q ds ≤ Cnx q . By combining the above estimates we obtain that where the last inequality follows from the assumption γ > −1/β. For s ∈ (−∞, 0) we use the assumption (1.6) on g to obtain and for α < 1 + 1/q we have that where we used 1 < α < 1 + 1/q in the last inequality. The above estimates imply for α = 1 + 1/q that   where the last inequality follows since α > 1. For α = 1 + 1/q, assumption (1.6) is satisfied forα = α − for all > 0 small enough. Hence, by (4.18) used withα we obtain that (4.19) is bounded by Cn 1−q/2 by choosing small enough.
The assumption that g(x) = 0 for all x < 0, implies that A n (x, s) = 0 for all s > n, and hence (4.17) and (4.20) show that Next, we treat the term I 3 . For the integral we need to distinguish different cases, namely s ≤ −x 1/α , −x 1/α < s ≤ n − x 1/α − 1 and n − x 1/α − 1 < s ≤ n.
We start with the case s ≤ −x 1/α . Note that and observe that x √ n(−s) −α > 1 if and only if s > −x 1/α n 1/(2α) . We obtain the inequality On the other hand, we have that

By (4.22) we thus conclude that
(4.23) The substitution v = x −1/α (n − s) yields where we used the assumption αβ > 2 in the second inequality. Finally, for the last case n − x 1/α − 1 < s ≤ n we have Summarizing, we arrive at the bound Next, we will bound the term I 2 as follows: where we recall that A n (x, s) = 0 for s > n.
We note that x 1/α n −1/2 ≤ 1 if and only if x ≤ n α/2 , and write J 1 as J 1 = J 1 + J 1 . Note where we have used (4.27) in the first inequality. Furthermore, where we have applied (4.27) in the first inequality, and the substitution v = n −1 x 1/α in the second equality.
Lemma 4.7. There exists a constant C > 0 such that Proof. Applying the inequality (4.4) we conclude that which together with Lemma 4.6 implies (4.32).

Lemma 4.8. There exists a finite constant C such that
A n (z) 2 λ(dz) ≤ C, where the second inequality follows from the substitution u = x 2 g(t 1 − s)g(t 2 − s), and the last inequality is a consequence of the assumption that αβ > 2.
Our proof of (4.34) relies on the estimate Moreover, we have , (4.36) where the equality follows by the substitution u = x    where the last inequality follows by Lemma 4.6. Lemma 4.2 of [17] shows that Hence, a combination of (4.39), the inequality |D z V n | ≤ A n (z), cf.  The two inequalities (4.38) and (4.40) yield the bound (4.37), which completes the proof of the lemma.