Bounds for non-Gaussian approximations of U-statistics

Yuri V.Borovskikh^{∗}  Neville C. Weber^{†}

16 August, 2000

Running Head: Bounds for U-statistics.

MSC 2000 classification. Primary 60F05; secondary 62E20.

Mailing Address: N. C. Weber School of Mathematics and Statistics, F07 University of Sydney N.S.W. 2006 AUSTRALIA Telephone: +61 2 9351 4249.

∗Department of Applied Mathematics, Transport University, Moskovsky Avenue, 9, 190031, St.Petersburg, Russia. Email:borovski@spb.cityline.ru

†School of Mathematics and Statistics, F07, University of Sydney, N.S.W. 2006, Australia. Email: neville@maths.usyd.edu.au

Abstract

Let X, X̄, X_{1}, …, X_{n} be i.i.d. random variables taking values in a measurable space (𝔛, 𝔅). Consider U-statistics of degree two

U_{n} = n^{−1} ∑_{1≤i<j≤n} Φ(X_{i}, X_{j}),

with symmetric, degenerate kernel Φ : 𝔛^{2} ↦ ℝ. Let

U_{∞} = \frac{1}{2} ∑_{j=1}^{∞} q_{j}(τ_{j}^{2} − 1),

where {q_{j}} are eigenvalues of the Hilbert-Schmidt operator associated with the kernel Φ and {τ_{j}} are i.i.d. standard normal random variables. If 𝔼Φ^{2}(X, X̄) < ∞ then

Δ_{n} := sup_{x} |ℙ{U_{n} ≤ x} − ℙ{U_{∞} ≤ x}| → 0 as n → ∞.

Upper bounds for Δ_{n} are established under the moment condition 𝔼Φ^{2}(X, X̄) = 1, provided that at least thirteen eigenvalues of the operator do not vanish. In Theorem 1.1 the bound is expressed via terms containing tail and truncated moments. The proof is based on the method developed by Bentkus and Götze (1999).

Key words and phrases. U-statistics, U-statistical sums, symmetric degenerate kernel, Gaussian random variables, tail moments.

1 Introduction

Let X, X̄, X_{1}, …, X_{n} be independent and identically distributed (i.i.d.) random variables taking values in an arbitrary measurable space (𝔛, 𝔅). Let Φ : 𝔛^{2} ↦ ℝ denote a measurable symmetric function, that is, Φ(x, y) = Φ(y, x), for all x, y ∈ 𝔛. Consider U-statistics of degree two

U_{n} = n^{−1} ∑_{1≤i<j≤n} Φ(X_{i}, X_{j}).  (1.1)

In what follows we assume σ^{2} = 𝔼Φ^{2}(X, X̄) < ∞ and that the kernel Φ is degenerate, that is, 𝔼Φ(x, X) = 0 for all x ∈ 𝔛. The variance of U_{n} is (n − 1)σ^{2}/(2n).

Let {q_{j}} denote the eigenvalues of the Hilbert-Schmidt operator, ℚ, associated with the kernel Φ,

ℚf(x) = ∫ ; 𝔛 Φ(x, y)f(y)P (dy) = 𝔼Φ(x, X)f(X), f ∈ L^{2},  (1.2)

where L^{2} = L^{2}(𝔛, 𝔅, P) denotes the Hilbert space of square integrable real functions defined on the measurable space (𝔛, 𝔅, P) with measure P = L(X). Further

𝔼Φ^{2}(X, X̄) = ∑_{j=1}^{∞} q_{j}^{2}.

Without loss of generality we shall assume throughout that |q_{1}| ≥ |q_{2}| ≥ …. Let {e_{j}} denote a corresponding orthogonal, complete system of eigenfunctions of the operator ℚ. Then in L^{2}(𝔛^{2}, 𝔅^{2}, P × P)

Φ(x, y) = ∑_{j=1}^{∞} q_{j}e_{j}(x)e_{j}(y),  (1.3)

since the kernel Φ is degenerate.

Let {τ_{j}} be a sequence of i.i.d. standard normal random variables. The asymptotic distribution of U_{n} is non-Gaussian and is the same as the distribution of the random variable

U_{∞} = \frac{1}{2} ∑_{j=1}^{∞} q_{j}(τ_{j}^{2} − 1).  (1.4)

Introduce

Δ_{n} = sup_{x} |ℙ{U_{n} ≤ x} − ℙ{U_{∞} ≤ x}|.  (1.5)

Our basic aim is to establish upper bounds for Δ_{n} under the minimal moment condition 𝔼Φ^{2}(X, X̄) < ∞ provided at least q_{13} ≠ 0. Without loss of generality we may assume that σ = 1.

To formulate our results we need the following definitions and notation. Define

∥x∥ = (𝔼Φ^{2}(X, x))^{1/2}, x ∈ 𝔛  (1.6)

and the degenerate, truncated kernel

h(x, y) = ∫ ; 𝔛 ∫ ; 𝔛 Φ(z, z̄)I(|Φ(z, z̄)| ≤ n)I(∥z∥ ≤ \sqrt{n})I(∥z̄∥ ≤ \sqrt{n})

×(δ_{x}(dz) − P (dz))(δ_{y}(dz̄) − P (dz̄)), x, y ∈ 𝔛,  (1.7)

where I(·) denotes the indicator function and δ_{x}(·) is Dirac’s δ-measure. Introduce the tail and truncated moments

α_{n} = 𝔼∥X∥^{2}I(∥X∥ > \sqrt{n}), β_{n} = 𝔼Φ^{2}(X, X̄)I(|Φ(X, X̄)| > n),
γ_{n} = n^{−1}𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{n}), λ_{n} = n^{−2}𝔼Φ^{4}(X, X̄)I(|Φ(X, X̄)| ≤ n),
μ_{n} = n^{−1}|𝔼Φ^{3}(X, X̄)I(|Φ(X, X̄)| ≤ n)|

and let

L_{n} = α_{n} + β_{n} + γ_{n} + λ_{n}, n ≥ 1.

Note that L_{n} → 0 as n → ∞ and since 𝔼∥X∥^{2} = 1

n^{−1} ≤ 2(n^{−1}α_{n}^{2} + γ_{n}) and n^{−2} ≤ 8(n^{−2}α_{n}^{4} + γ_{n}^{2}).  (1.8)

Further denote

p = exp{−(16s)^{3}/|q_{s}|},  (1.9)

provided that q_{s} ≠ 0, s ≥ 1, and define an integer m(s, q_{s}) ≥ 1 by the formula

m(s, q_{s}) = min{m ≥ 1 : L_{m} ≤ 2^{−15}s^{−3}p}.

In what follows we use c and c(·), with or without indices, to denote absolute, generic, positive constants and constants depending on the parameters in the brackets. In particular, c(q_{s}) = exp{c/|q_{s}|} if q_{s} ≠ 0.

Theorem 1.1. If q_{13} ≠ 0 then

Δ_{n} ≤ exp{c/|q_{13}|}(L_{n} + μ_{n} + n^{−1}m(13, q_{13})).  (1.10)

Throughout we shall assume that

L_{n} ≤ p/8.  (1.11)

Since Δ_{n} ≤ 1, if L_{n} ≥ p/8 then Δ_{n} ≤ 8p^{−1}L_{n} ≤ c(q_{13})L_{n}. That is (1.10) holds. Similarly we may assume that np > m(13, q_{13}).

Under additional moment conditions the estimate (1.10) implies a bound of order O(n^{−1}).

Corollary 1.1. Let a := 𝔼∥X∥^{4} < ∞ and assume

b := sup_{x}(x^{3}ℙ(|Φ(X, X̄)| > x)) < ∞.

Then

α_{n} ≤ n^{−1}a,  β_{n} ≤ 3n^{−1}b,  γ_{n} ≤ n^{−1}a,  λ_{n} ≤ 4n^{−1}b

and

m(13, q_{13}) ≤ 2^{15}13^{3}(2a + 7b)p^{−1} + 1.

If

|𝔼Φ^{3}(X, X̄)I(|Φ(X, X̄)| ≤ n)| ≤ d < ∞  (1.12)

then

Δ_{n} ≤ c(q_{13}) (a + b + d) n^{−1}.  (1.13)

Note that if 𝔼|Φ|^{3} < ∞ then the condition (1.12) is satisfied with d = 𝔼|Φ|^{3} and moreover b ≤ 𝔼|Φ|^{3}. In this case the Corollary 1.1 is reduced to the result by Bentkus and Götze (1999)

Δ_{n} ≤ c(q_{13})(𝔼∥X∥^{4} + 𝔼|Φ(X, X̄)|^{3})n^{−1}.

However the conditions b < ∞ and d < ∞ can satisfied but nevertheless 𝔼|Φ|^{3} = ∞, as shown in the following example.

Example 1.1. Let X be a random variable with the uniform distribution on 𝔛 = [−1, 1]. For x, y ∈ 𝔛 introduce the functions

ϕ(x, y) = (x + y)|x − y|^{−1/3},

ϕ_{1}(x) = 𝔼ϕ(x, X) = \frac{3}{10}(4x − 1)(1 + x)^{2/3} + \frac{3}{10}(4x + 1)(1 − x)^{2/3}.

It is obvious that

𝔼ϕ(X, X̄) = 0, ϕ(x, y) = ϕ(y, x), ϕ(x, y) = −ϕ(−x, −y), x, y ∈ 𝔛.

Define the degenerate kernel Φ(x, y) = ϕ(x, y) − ϕ_{1}(x) − ϕ_{1}(y). Then

Φ(x, y) = Φ(y, x), Φ(x, y) = −Φ(−x, −y), x, y ∈ 𝔛,

and therefore for this kernel 𝔼Φ^{3}(X, X̄)I(|Φ(X, X̄)| ≤ n) = 0. Simple calculations show that

sup_{x}(x^{3}ℙ(|Φ(X, X̄)| > x)) ≤ 2^{16},

∥x∥ = (𝔼Φ^{2}(x, X))^{1/2} ≤ 12, x ∈ 𝔛

but

𝔼|Φ(X, X̄)|^{3} = ∞.

Consequently, in Corollary 1.1 we obtain the optimal bound of order O(n^{−1}) in (1.13) even though the third absolute moment of the kernel Φ is not finite. Thus 𝔼|Φ|^{3} < ∞ is not necessary to achieve a bound of order O(n^{−1}) for Δ_{n}.

In the sequel let ℝ^{∞} be the space of all real sequences x = (x_{1}, x_{2}, …), x_{j} ∈ ℝ. Let l_{2} ⊂ ℝ^{∞} be the Hilbert space consisting of x ∈ ℝ^{∞}, such that

|x|^{2} = (x, x), |x| < ∞, (x, y) = ∑_{j=1}^{∞} x_{j}y_{j}.

Define ℚx = (q_{1}x_{1}, q_{2}x_{2}, …) with x ∈ ℝ^{∞}. The random eigenfunctions vector e(X) = (e_{1}(X), e_{2}(X), …) takes values in ℝ^{∞}. This vector has mean zero and identity covariance matrix, since {e_{j}(X)} is a system of uncorrelated random variables with unit variances. According to (1.3) we can write

Φ(X, X̄) = (ℚe(X), e(X̄)).  (1.14)

Equation (1.14) enables us to assume that the measurable space 𝔛 is ℝ^{∞} and the random variable X is a random vector taking values in ℝ^{∞}, with mean zero and identity covariance. Also without loss of generality we can assume that the kernel Φ(x, y) is linear function in each of its arguments.

Introduce the Gaussian random vector G = (τ_{1}, τ_{2}, …) taking values in ℝ^{∞}, where τ_{1}, τ_{2}, … denote i.i.d. standard normal random variables. In this case in (1.14) e(G) = G and e_{j}(G) = τ_{j}, j = 1, 2, …. The random vectors G and X have mean zero and equal covariances, that is

𝔼Φ(X, x) = 𝔼Φ(G, x) = 0,
𝔼Φ(X, x)Φ(X, y) = 𝔼Φ(G, x)Φ(G, y).  (1.15)

The paper is organized as follows. Lemma 2.1 in Section 2 provides a truncation inequality for U-statistical type sums which enables us to introduce simultaneously truncated random variables and truncated kernels. In Lemma 2.2 the eigenvalues associated with the original and truncated kernels are compared. Section 3 contains the non-degeneracy condition of Bentkus and Götze (1999) for the non-Gaussian random vector under the minimal moment condition σ = 1. The fourth section provides bounds for associated characteristic functions and in Section 5 we prove Theorem 1.1 by applying Lemma 4.2 of Bentkus and Götze (1999).

2 Double truncation

Introduce the truncated functions

f(x, y) = Φ(x, y)I(∥x∥ ≤ \sqrt{n})I(∥y∥ ≤ \sqrt{n}), g(x, y) = f(x, y)I(|Φ(x, y)| ≤ n),

x, y ∈ 𝔛, and the truncated random variables X_{j}^{tr} = X_{j}I(∥X_{j}∥ ≤ \sqrt{n}), j = 1, …, n. Let s(Φ) = s(Φ; X_{1}, …, X_{n}), where

s(Φ; X_{1}, …, X_{n}) = ∑_{1≤i<j≤n} Φ(X_{i}, X_{j}).

The following lemma contains the double truncation inequality, which enables us to introduce X_{j}^{tr} instead of X_{j}, j = 1, …, n and the random variables g(X_{i}, X_{j}) instead of Φ(X_{i}, X_{j}), 1 ≤ i < j ≤ n.

Lemma 2.1. For all x ∈ ℝ

|ℙ(s(Φ) ≤ x) − ℙ(s(g) ≤ x)|
≤ n ℙ(∥X∥ > \sqrt{n}) + \frac{1}{2}n(n − 1)ℙ(|Φ(X, X̄)| > n).  (2.1)

Proof. First we shall show that for x ∈ ℝ

|ℙ(s(Φ) ≤ x) − ℙ(s(f) ≤ x)| ≤ n ℙ(∥X∥ > \sqrt{n}).  (2.2)

Consider the events

A_{j} = \{∥X_{j}∥ ≤ \sqrt{n}\}, A_{j}^{c} = \{∥X_{j}∥ > \sqrt{n}\}, j = 1, …, n,

A = ⋂^{n} ; j=1A_{j}, A^{c} = ⋃^{n} ; j=1A_{j}^{c},

where A^{c} denotes the complement of A.

Note that ℙ(A^{c}) ≤ n ℙ(∥X∥ > \sqrt{n}). Taking account of the linearity of Φ(x, y) in each of its arguments we obtain

ℙ({s(Φ; X_{1}, …, X_{n}) ≤ x} ∩ A) = ℙ({s(Φ; X_{1}^{tr}, …, X_{n}^{tr}) ≤ x} ∩ A)

= ℙ({s(f; X_{1}, …, X_{n}) ≤ x} ∩ A).

Hence

ℙ(s(Φ) ≤ x) = ℙ({s(f) ≤ x} ∩ A) + ℙ({s(Φ) ≤ x} ∩ A^{c})
≤ ℙ(s(f) ≤ x) + ℙ(A^{c}).

Similarly

ℙ(s(f) ≤ x) ≤ ℙ(s(Φ) ≤ x) + ℙ(A^{c}).

This proves (2.2). Next we shall show that for x ∈ ℝ

|ℙ(s(f) ≤ x) − ℙ(s(g) ≤ x)| ≤ \frac{1}{2}n(n − 1)ℙ(|Φ(X, X̄)| > n).  (2.3)

Consider the events

A_{ij} = {|f(X_{i}, X_{j})| ≤ n}, 1 ≤ i < j ≤ n, A = ⋂_{1≤i<j≤n}A_{ij}, A^{c} = ⋃_{1≤i<j≤n}A_{ij}^{c}.

Here ℙ(A^{c}) ≤ 2^{−1}n(n − 1)ℙ(|Φ(X, X̄)| > n). Further we write

ℙ(s(f) ≤ x) = ℙ({s(g) ≤ x} ∩ A) + ℙ({s(f) ≤ x} ∩ A^{c})

≤ ℙ(s(g) ≤ x) + ℙ(A^{c}).

Similarly

ℙ(s(g) ≤ x) ≤ ℙ(s(f) ≤ x) + ℙ(A^{c}).

This proves (2.3) and the Lemma follows.

To complete this section we shall compare the eigenvalues of the Hilbert-Schmidt operators associated with the initial kernel Φ and the truncated kernel h defined at (1.7). Introduce the Hilbert-Schmidt operator ℚ̄ : L^{2} ↦ L^{2}, associated with the kernel h,

ℚ̄f(x) = ∫ ; 𝔛 h(x, y)f(y)P (dy) = 𝔼h(x, X)f(X), f ∈ L^{2}.  (2.4)

Let {q̄_{j}} denote the eigenvalues of ℚ̄. Without loss of generality assume that |q̄_{1}| ≥ |q̄_{2}| ≥ …. Further let {ē_{j}} denote an orthonormal, complete system of eigenfunctions of ℚ̄ ordered by decreasing absolute values of the corresponding eigenvalues q̄_{1}, q̄_{2}, …. Since the kernel h is degenerate

h(x, y) = ∑_{j=1}^{∞} q̄_{j}ē_{j}(x)ē_{j}(y)  (2.5)

in L^{2}(𝔛^{2}, 𝔅^{2}, P × P). The following lemma contains lower and upper bounds for q̄_{j} in terms of q_{j}.

Lemma 2.2. If q_{s} ≠ 0 for some s ≥ 1 then

\frac{1}{2}|q_{j}| ≤ |q̄_{j}| ≤ 2|q_{j}|, for 1 ≤ j ≤ s  (2.6)

provided that α_{n} + β_{n} ≤ p.

Proof. Define the symmetric kernel

Φ_{1}(x, y) = ∫ ; 𝔛 Φ(x, z)Φ(y, z)P (dz).  (2.7)

According to (1.2) for f ∈ L^{2}

ℚ^{2}f(x) = ℚ(ℚf)(x) = ℚ(∫ ; 𝔛 Φ(x, y)f(y)P (dy))

= ∫ ; 𝔛 Φ(x, z)(∫ ; 𝔛 Φ(z, y)f(y)P (dy))P (dz)

= ∫ ; 𝔛 Φ_{1}(x, y)f(y)P (dy).

Hence ℚ^{2} : L^{2} ↦ L^{2} is the Hilbert-Schmidt operator, associated with the kernel Φ_{1}. Further for any f ∈ L^{2}

(ℚ^{2}f, f) = ∫ ; 𝔛 {∫ ; 𝔛 Φ(x, z)f(x)P (dx)}^{2}P (dz),  (2.8)

that is, the operator ℚ^{2} is positive definite. From (1.3)

∫ ; 𝔛 Φ_{1}(x, y)e_{j}(y)P (dy) = q_{j}^{2}e_{j}(x), j ≥ 1  (2.9)

so that {e_{j}} and {q_{j}^{2}} are the orthonormal complete system of eigenfunctions and corresponding eigenvalues of the operator ℚ^{2}. Therefore by the theory of Schmidt-Courant (see, for example, Dunford and Schwartz (1961), Chapter 10)

q_{1}^{2} = sup{(ℚ^{2}f, f) : |f| = 1},  (2.10)

q_{j}^{2} = inf{sup{(ℚ^{2}f, f) : |f| = 1, (f, f_{i}) = 0} : f_{i} ∈ L^{2}, i = 1, …, j − 1}, j ≥ 2.

Here the supremum is taken for fixed f_{i} and then the infimum is taken over all possible f_{i} ∈ L^{2}, i = 1 …, j − 1.

Further define the symmetric kernel

h_{1}(x, y) = ∫ ; 𝔛 h(x, z) h(y, z) P (dz).  (2.11)

and the Hilbert-Schmidt operator ℚ̄^{2} : L^{2} ↦ L^{2}, associated with the kernel h_{1}. The eigenfunctions and the eigenvalues of ℚ̄^{2} are {ē_{j}} and {q̄_{j}^{2}} respectively. A similar relation to (2.10) holds for the operator ℚ̄^{2} and the eigenvalues {q̄_{j}^{2}}. Hence, for all j ≥ 1, we can write

|q_{j}^{2} − q̄_{j}^{2}| ≤ sup{|((ℚ^{2} − ℚ̄^{2})f, f)| : |f| = 1}.  (2.12)

By the definition of the operators ℚ^{2} and ℚ̄^{2}

((ℚ^{2} − ℚ̄^{2})f, f) = ∫ ; 𝔛 ({∫ ; 𝔛 Φ(x, y)f(x)P (dx)}^{2} − {∫ ; 𝔛 h(x, y)f(x)P (dx)}^{2})P (dy).

(2.13)

From (1.7)

h(x, y) = Φ(x, y)I(|Φ(x, y)| ≤ n)I(∥x∥ ≤ \sqrt{n})I(∥y∥ ≤ \sqrt{n})  (2.14)
−𝔼Φ(x, X̄)I(|Φ(x, X̄)| ≤ n)I(∥x∥ ≤ \sqrt{n})I(∥X̄∥ ≤ \sqrt{n})
−𝔼Φ(X, y)I(|Φ(X, y)| ≤ n)I(∥X∥ ≤ \sqrt{n})I(∥y∥ ≤ \sqrt{n})
+𝔼Φ(X, X̄)I(|Φ(X, X̄)| ≤ n)I(∥X∥ ≤ \sqrt{n})I(∥X̄∥ ≤ \sqrt{n}).

Hence

h(x, y) = Φ(x, y) + ϕ(x, y)  (2.15)

with the representation

ϕ(x, y) = θ(x, y) − 𝔼θ(x, X̄) − 𝔼θ(X, y) + 𝔼θ(X, X̄),  (2.16)

where

θ(x, y) = −Φ(x, y)I(∥x∥ > \sqrt{n}) − Φ(x, y)I(∥y∥ > \sqrt{n})  (2.17)
+Φ(x, y)I(∥x∥ > \sqrt{n})I(∥y∥ > \sqrt{n})
−Φ(x, y)I(|Φ(x, y)| > n)I(∥x∥ ≤ \sqrt{n})I(∥y∥ ≤ \sqrt{n}).

Simple calculations show that

𝔼ϕ^{2}(X, X̄) = 𝔼θ^{2}(X, X̄) + 3(𝔼θ(X, X̄))^{2},  (2.18)

and,

𝔼θ^{2}(X, X̄) ≤ 12α_{n} + 4β_{n} and |𝔼θ(X, X̄)| ≤ \frac{1}{n}(α_{n} + β_{n}).  (2.19)

Then

𝔼ϕ^{2}(X, X̄) ≤ 18α_{n} + 10β_{n}.  (2.20)

By substituting (2.15) in (2.13) and using the Hölder’s inequality and (2.20) we obtain

sup{|((ℚ^{2} − ℚ̄^{2})f, f)| : |f| = 1} ≤ 2(18α_{n} + 10β_{n})^{1/2} + 18α_{n} + 10β_{n}. (2.21)

The condition α_{n} + β_{n} ≤ p of this lemma is valid by assumption (1.11), since α_{n} + β_{n} ≤ L_{n}. Therefore from (2.12) and (2.21) we have for all j ≥ 1

|q_{j}^{2} − q̄_{j}^{2}| ≤ 40\sqrt{p}.  (2.22)

If j = 1, …, s then |q_{j}| ≥ |q_{s}| > 0 and

p = exp{−(16s)^{3}/|q_{s}|} ≤ exp{−(16s)^{3}/|q_{j}|} ≤ 16^{−4}q_{j}^{4}.

From here and (2.22) we get (2.6).

As in (2.15) we can define the degenerate kernel h̄(x, y) and write

h̄(x, y) = Φ(x, y) + ϕ̄(x, y)  (2.23)

truncating at m instead of n. Using (2.5) we can assume throughout that h(x, y) and h̄(x, y) are linear in their arguments x, y ∈ ℝ^{∞}.

3 Non-degeneracy condition

For a matrix 𝔸 = (a_{ij}), 1 ≤ i, j ≤ s, we denote by |𝔸|_{∞} = max{|a_{ij}| : 1 ≤ i, j ≤ s}. Define the random matrix 𝔸(z, h) = (a_{ij}(h)), 1 ≤ i, j ≤ s, with entries a_{ij}(h) = h(z_{i}, z̄_{j}) where z_{i} and z̄_{j} are independent copies of z. Following Bentkus and Götze (1999) we say that z and h satisfy the non-degeneracy condition N(p, δ, s, z, h) if

ℙ{|𝔸(z, h) − 𝕀|_{∞} ≤ δ} ≥ p,

where 𝕀 denotes the identity matrix, 0 < p ≤ 1, δ ≥ 0 and s ≥ 1. From their Lemma 5.3 for a Gaussian random vector G and q̄_{s} ≠ 0 for some s ≥ 1 the condition N(2p, δ, s, ϵG, h) is fulfilled with p = exp{−c_{s}/(ϵ^{2}|q̄_{s}|)}, provided that δ ≤ 1 and 4sσ̄ϵ^{2} ≤ δ, where σ̄^{2} = 𝔼h^{2}(X, X̄) and c_{s} is a positive constant. Their inequality (5.25) gives the constant

c_{1}(s) = 1/(2^{s/2}Γ(1 + s/2)) ≥ (s + 2)^{−(s+1)/2} exp(s/2),

and moreover c_{s} = 8s. Since σ = 1, using (2.14) we have σ̄ ≤ 4. Hence taking into account (2.6) we have

ℙ{|𝔸(ϵG, h) − 𝕀|_{∞} ≤ δ} ≥ 2p  (3.1)

with p = exp{−16s/(ϵ^{2}|q_{s}|)} for δ ≤ 1 and 16sϵ^{2} ≤ δ. Set ϵ = δ = 1/(16s). Then in (3.1) p = exp{−(16s)^{3}/(|q_{s}|)}.

The following lemma provides conditions for the non-degeneracy of the non-Gaussian random vectors under the minimal moment condition 𝔼Φ^{2}(X, X̄) = 1. Denote

Y = (2m)^{−1/2}(X̃_{1} + … + X̃_{m}), m ≥ 1  (3.2)

and let Ȳ = (2m)^{−1/2}( ˜X̄_{1}+…+ ˜X̄_{m}) be a independent copy of Y. Here X̃_{k} = X_{k}−X_{k}^{1}, ˜X̄_{l} = X̄_{l} − X̄_{l}^{1}, 1 ≤ k, l ≤ m, are independent symmetrizations and X_{k}, X_{k}^{1}, X̄_{l}, X̄_{l}^{1}, 1 ≤ k, l ≤ m are independent copies of X. Hence we can write

h(Y, Ȳ) = \frac{1}{2m} ∑_{k=1}^{m} ∑_{l=1}^{m} h(X̃_{k}, ˜ X̄_{l}).  (3.3)

Further let Y_{1}, …, Y_{s} and Ȳ_{1}, …, Ȳ_{s} be independent copies of Y, that is,

Y_{i} = (2m)^{−1/2} ∑_{k=1}^{m} X̃_{ik}, Ȳ_{j} = (2m)^{−1/2} ∑_{l=1}^{m} ˜ X̄_{jl},

with the symmetrizations X̃_{ik} = X_{ik} − X_{ik}^{1}, ˜ X̄_{jl} = X̄_{jl} − X̄_{jl}^{1}, where X_{ik}, X̄_{jl} and X_{ik}^{1}, X̄_{jl}^{1}, 1 ≤ k, l ≤ m, are independent copies of the sequence X_{i}, X̄_{j}, 1 ≤ i, j ≤ s. Therefore by (3.3) the entries a_{ij}(h) of the random matrix 𝔸(ϵY, h) can be written as

a_{ij}(h) = \frac{ϵ^{2}}{2m} ∑_{k=1}^{m} ∑_{l=1}^{m} h(X̃_{ik}, ˜ X̄_{jl}).  (3.4)

A similar expression holds for the entries a_{ij}(h̄) of the random matrix 𝔸(ϵY, h̄). Lemma 3.1. Let q_{s} ≠ 0 for some s ≥ 1 and

ϵ = δ = \frac{1}{16s}, p = exp\{−16s/(ϵ^{2}|q_{s}|)\}.  (3.5)

Then the condition N(2p, δ, s, ϵG, h) implies the condition N(p, 8δ, s, ϵY, h) provided that m ≥ m(s, q_{s}), where

m(s, q_{s}) = min{m ≥ 1 : L_{m} ≤ 2^{−15}s^{−3}p}  (3.6)

and L_{m} = α_{m} + β_{m} + γ_{m} + λ_{m}.

Proof. We shall follow Bentkus and Götze (1999). It suffices to show that

ℙ{|𝔸(ϵY, h) − 𝕀|_{∞} ≤ 8δ} ≥ p  (3.7)

for m ≥ m(s, q_{s}). First we shall show that for m ≥ 1

ℙ{|𝔸(ϵY, h) − 𝕀|_{∞} ≤ 8δ}  (3.8)
≥ ℙ\{|𝔸(ϵY, h̄) − 𝕀|_{∞} ≤ 4δ\} − \frac{9}{16^{2}} (α_{m} + β_{m}) − \frac{9}{16^{2}} (α_{n} + β_{n}).

Using (2.15), (2.23) and (3.4) we write 𝔸(ϵY, h) = 𝔸(ϵY, h̄) + 𝔸(ϵY, ϕ − ϕ̄) and

|𝔸(ϵY, h) − 𝕀|_{∞} ≤ |𝔸(ϵY, h̄) − 𝕀|_{∞} + |𝔸(ϵY, ϕ − ϕ̄)|_{∞}.  (3.9)

The random variables a_{ij}(ϕ − ϕ̄) are identically distributed and

ℙ\{|𝔸(ϵY, ϕ − ϕ̄)|_{∞} ≥ 4δ\} ≤ \frac{1}{16δ^{2}} 𝔼 max_{1≤i,j≤s} a_{ij}^{2}(ϕ − ϕ̄)  (3.10)
≤ \frac{s^{2}}{16δ^{2}} 𝔼a_{11}^{2}(ϕ − ϕ̄)
≤ \frac{s^{2}}{8δ^{2}} (𝔼a_{11}^{2}(ϕ) + 𝔼a_{11}^{2}(ϕ̄))
≤ \frac{s^{2}}{8δ^{2}} ϵ^{4}(𝔼ϕ^{2}(X, X̄) + 𝔼ϕ̄^{2}(X, X̄))
≤ \frac{1}{8 · 16^{2}} (18α_{n} + 10β_{n} + 18α_{m} + 10β_{m}), using (2.21)
≤ \frac{9}{16^{2}} (α_{m} + β_{m}) + \frac{9}{16^{2}} (α_{n} + β_{n}).

Further applying these results we have

ℙ{|𝔸(ϵY, h) − 𝕀|_{∞} ≤ 8δ}  (3.11)
≥ ℙ{|𝔸(ϵY, h̄) − 𝕀|_{∞} + |𝔸(ϵY, ϕ − ϕ̄)|_{∞} ≤ 8δ}
≥ 𝔼I{|𝔸(ϵY, h̄) − 𝕀|_{∞} + |𝔸(ϵY, ϕ − ϕ̄)|_{∞} ≤ 8δ}I{|𝔸(ϵY, ϕ − ϕ̄)|_{∞} ≤ 4δ}
≥ 𝔼I{|𝔸(ϵY, h̄) − 𝕀|_{∞} ≤ 4δ}I{|𝔸(ϵY, ϕ − ϕ̄)|_{∞} ≤ 4δ}
≥ ℙ{|𝔸(ϵY, h̄) − 𝕀|_{∞} ≤ 4δ} − ℙ{|𝔸(ϵY, ϕ − ϕ̄)|_{∞} ≥ 4δ}
≥ ℙ\{|𝔸(ϵY, h̄) − 𝕀|_{∞} ≤ 4δ\} − \frac{9}{16^{2}} (α_{m} + β_{m}) − \frac{9}{16^{2}} (α_{n} + β_{n}).

This proves (3.8).

Next we shall show that for m ≥ 1

ℙ{|𝔸(ϵY, h̄)−𝕀|_{∞} ≤ 4δ} ≥ ℙ{|𝔸(ϵG, h̄)−𝕀|_{∞} ≤ 2δ}−3·4^{3}·s^{3}λ_{m}−15·4^{4}·s^{3}γ_{m}. (3.12)

Introduce the four times continuously differentiable function φ : ℝ ↦ ℝ with bounded derivatives such that φ(x) = 1, for |x| ≤ 1, φ(x) = 0, for |x| ≥ 2, and

φ(x) = (1 − (\frac{1}{3}x^{2} − \frac{1}{3})^{5})^{5}, for 1 ≤ |x| ≤ 2.

It is obvious that |φ|_{∞} = 1, where |φ|_{∞} = sup_{x} |φ(x)|. The first four derivatives of φ(x) for 1 ≤ |x| ≤ 2 have the form

φ^{(1)}(x) = −\frac{50}{3}x\{(\frac{1}{3}x^{2} − \frac{1}{3}) − (\frac{1}{3}x^{2} − \frac{1}{3})^{6}\}^{4},

φ^{(2)}(x) = −\frac{50}{3}\{(\frac{1}{3}x^{2} − \frac{1}{3}) − (\frac{1}{3}x^{2} − \frac{1}{3})^{6}\}^{3}

×\{\frac{8}{3} + 9(\frac{1}{3}x^{2} − \frac{1}{3}) − 16(\frac{1}{3}x^{2} − \frac{1}{3})^{5} − 49(\frac{1}{3}x^{2} − \frac{1}{3})^{6}\},

φ^{(3)}(x) = −\frac{50}{3}x^{2}\{(\frac{1}{3}x^{2} − \frac{1}{3}) − (\frac{1}{3}x^{2} − \frac{1}{3})^{6}\}^{2}φ_{1}(x),
φ^{(4)}(x) = −\frac{50}{3}\{(\frac{1}{3}x^{2} − \frac{1}{3}) − (\frac{1}{3}x^{2} − \frac{1}{3})^{6}\}

×\{(\frac{2}{3} − 4(\frac{1}{3}x^{2} − \frac{1}{3})^{5})φ_{1}(x) + ((\frac{1}{3}x^{2} − \frac{1}{3}) − (\frac{1}{3}x^{2} − \frac{1}{3})^{6})φ_{2}(x)\},

where

φ_{1}(x) = \frac{16}{3} + 18(\frac{1}{3}x^{2} − \frac{1}{3}) − 64(\frac{1}{3}x^{2} − \frac{1}{3})^{5} − 206(\frac{1}{3}x^{2} − \frac{1}{3})^{6}

+192(\frac{1}{3}x^{2} − \frac{1}{3})^{10} + 588(\frac{1}{3}x^{2} − \frac{1}{3})^{11},

φ_{2}(x) = \frac{52}{3} + 54(\frac{1}{3}x^{2} − \frac{1}{3}) − \frac{640}{3}(\frac{1}{3}x^{2} − \frac{1}{3})^{4} − 1528(\frac{1}{3}x^{2} − \frac{1}{3})^{5}

−2678(\frac{1}{3}x^{2} − \frac{1}{3})^{6} + 1280(\frac{1}{3}x^{2} − \frac{1}{3})^{9} + 6424(\frac{1}{3}x^{2} − \frac{1}{3})^{10}

+13524(\frac{1}{3}x^{2} − \frac{1}{3})^{11}.

Note that φ^{(k)}(x) = 0, k = 1, …, 4 for |x| ≤ 1 and |x| ≥ 2. Thus

|φ^{(1)}|_{∞} ≤ 34, |φ^{(2)}|_{∞} ≤ 1309, |φ^{(3)}|_{∞} ≤ 7160, |φ^{(4)}|_{∞} ≤ 108719. (3.13)

From the definition of φ we can write

ℙ\{|𝔸(ϵY, h̄) − 𝕀|_{∞} ≤ 4δ\} ≥ 𝔼 ∏_{1≤i,j≤s} φ(\frac{ϵ^{2}h̄(Y_{i}, Ȳ_{j}) − δ_{ij}}{2δ} )  (3.14)

= 𝔼 ∏_{1≤i,j≤s} φ(\frac{ϵ^{2}h̄(G_{i}, Ḡ_{j}) − δ_{ij}}{2δ} ) + R
≥ ℙ{|𝔸(ϵG, h̄) − 𝕀|_{∞} ≤ 2δ} − |R|,

where

R = 𝔼 ∏_{1≤i,j≤s} φ(\frac{ϵ^{2}h̄(Y_{i}, Ȳ_{j}) − δ_{ij}}{2δ} ) − 𝔼 ∏_{1≤i,j≤s} φ(\frac{ϵ^{2}h̄(G_{i}, Ḡ_{j}) − δ_{ij}}{2δ} ),

and δ_{ij} is the Kronecker delta. In what follows we shall show that

|R| ≤ 3 · 4^{3}s^{3}λ_{m} + 15 · 4^{4}s^{3}γ_{m}.  (3.15)

In R we shall replace the random vectors Y_{j} (or Ȳ_{j}) by corresponding Gaussian random vectors G_{j} (respectively, Ḡ_{j}) for 1 ≤ j ≤ s and then show that after 2s replacements the general error is bounded by the right hand side of (3.15). First we write the formula containing all 2s replacements

R = ∑_{l=1}^{s} 𝔼\{ ∏_{i=l+1}^{s} 𝔼_{Y} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(Y, Ȳ_{j}) − δ_{ij}}{2δ} ) ∏_{i=1}^{l−1} 𝔼_{G} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(G, Ȳ_{j}) − δ_{ij}}{2δ} )

×[𝔼_{Y} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(Y, Ȳ_{j}) − δ_{lj}}{2δ} ) − 𝔼_{G} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(G, Ȳ_{j}) − δ_{lj}}{2δ} )]\}

+ ∑_{l=1}^{s} 𝔼\{ ∏_{j=l+1}^{s} 𝔼Ȳ ∏_{i=1}^{s} φ(\frac{ϵ^{2}h̄(G_{i}, Ȳ) − δ_{ij}}{2δ} ) ∏_{j=1}^{l−1} 𝔼Ḡ ∏_{i=1}^{s} φ(\frac{ϵ^{2}h̄(G_{i}, Ḡ) − δ_{ij}}{2δ} )

×[𝔼Ȳ ∏_{i=1}^{s} φ(\frac{ϵ^{2}h̄(G_{i}, Ȳ) − δ_{il}}{2δ} ) − 𝔼Ḡ ∏_{i=1}^{s} φ(\frac{ϵ^{2}h̄(G_{i}, Ḡ) − δ_{il}}{2δ} )]\}.

Since |φ|_{∞} ≤ 1 then from this formula we get

|R| ≤ s(𝔼J(Ȳ) + 𝔼J(G)),  (3.16)

where

J(Ȳ) = |𝔼_{Y} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(Y, Ȳ_{j}) − δ_{1j}}{2δ} ) − 𝔼_{G} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(G, Ȳ_{j}) − δ_{1j}}{2δ} )|,

J(G) = |𝔼_{Y} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(Y, G_{j}) − δ_{1j}}{2δ} ) − 𝔼_{G} ∏_{j=1}^{s} φ(\frac{ϵ^{2}h̄(G, G_{j}) − δ_{1j}}{2δ} )|.

Now we shall estimate J(Ȳ). Fix ȳ_{j} = Ȳ_{j} for 1 ≤ j ≤ s. Introduce the random vectors

y = (ϵ^{2}h̄(Y, ȳ_{1}), …, ϵ^{2}h̄(Y, ȳ_{s})), g = (ϵ^{2}h̄(G, ȳ_{1}), …, ϵ^{2}h̄(G, ȳ_{s}))

and the Frechet differentiable function

ψ(x) = φ(\frac{x_{1} − 1}{2δ})φ(\frac{x_{2}}{2δ} ) ⋯ φ(\frac{x_{s}}{2δ} ), x = (x_{1}, …, x_{s}) ∈ ℝ^{s}.

We can rewrite J(ȳ) as

J(ȳ) = |𝔼ψ(y) − 𝔼ψ(g)|, ȳ = (ȳ_{1}, …, ȳ_{s}) ∈ ℝ^{s}.

Recall that Y =^{d} (X̃_{1} + ⋯ + X̃_{m})/\sqrt{2m} and we can write G, as

G =^{d} (G̃_{1} + ⋯ + G̃_{m})/\sqrt{2m}. Therefore

y d = ξ_{1} + ⋯ + ξ_{m}, with ξ_{j} = (ϵ^{2}h̄(X̃_{j}, ȳ_{l})/\sqrt{2m})_{1≤l≤s},

g d = ζ_{1} + ⋯ + ζ_{m}, with ζ_{j} = (ϵ^{2}h̄(G̃_{j}, ȳ_{l})/\sqrt{2m})_{1≤l≤s},

in addition the random vectors ξ_{1}, …, ξ_{m} (respectively, ζ_{1}, …, ζ_{m}) take values in ℝ^{s} and are independent and have identical, symmetric distributions. The Gaussian random vectors ζ_{j} have mean zero and their covariances are equal to those of ξ_{j}. Further, denoting

η_{j} = |𝔼ψ(s_{j} + ξ_{j}) − 𝔼ψ(s_{j} + ζ_{j})|, s_{j} = ζ_{1} + ⋯ + ζ_{j−1} + ξ_{j+1} + ⋯ + ξ_{m},

we get

J(ȳ) ≤ η_{1} + ⋯ + η_{m}.  (3.17)

Expanding into a Taylor series in powers of ξ_{j} and ζ_{j} and using symmetry and the equality of the covariances of ξ_{j} and ζ_{j} we obtain

η_{j} ≤ ∑_{ν_{1}+···+ν_{s}=4} \frac{1}{ν_{1}! ⋯ ν_{s}!}(𝔼|ξ_{j}|^{4} + 𝔼|ζ_{j}|^{4}) sup\{|(D^{ν}ψ)(x)| : x ∈ ℝ^{s}\} (3.18)

where

𝔼|ξ_{j}|^{4} = \frac{1}{(2m)^{2}} ϵ^{8}𝔼(∑_{l=1}^{s} h̄^{2}(X̃, ȳ_{l}))^{2}, 𝔼|ζ_{j}|^{4} = \frac{1}{(2m)^{2}} ϵ^{8}𝔼(∑_{l=1}^{s} h̄^{2}(G̃, ȳ_{l}))^{2},

and D^{ν}ψ is derivative of order ν = (ν_{1}, …, ν_{s}) of the function ψ, that is,

(D^{ν}ψ)(x) = \frac{∂^{ν_{1}}}{∂x_{1}^{ν_{1}}} φ(\frac{x_{1} − 1}{2δ}) \frac{∂^{ν_{2}}}{∂x_{2}^{ν_{2}}} φ(\frac{x_{2}}{2δ} ) ⋯ \frac{∂^{ν_{s}}}{∂x_{s}^{ν_{s}}} φ(\frac{x_{s}}{2δ} ).

For ν_{1} + ⋯ + ν_{s} = 4, ν_{j} ≥ 0, j = 1, …, s, under the condition |φ|_{∞} ≤ 1 we have

sup{|(D^{ν}ψ)(x)| : x ∈ ℝ^{s}}

≤ (2δ)^{−4} max{|φ^{(4)}|_{∞}, |φ^{(1)}|_{∞} · |φ^{(3)}|_{∞}, |φ^{(1)}|_{∞}^{2} · |φ^{(2)}|_{∞}, |φ^{(2)}|_{∞}^{2}, |φ^{(1)}|_{∞}^{4}}.

According to (3.13)

sup{|(D^{ν}ψ)(x)| : x ∈ ℝ^{s}} ≤ 17^{2} · 77^{2} · (2δ)^{−4}.  (3.19)

Further in (3.18)

∑_{ν_{1}+···+ν_{s}=4} \frac{1}{ν_{1}! ⋯ ν_{s}!} = \frac{1}{4!}s^{4}

and

𝔼|ξ_{j}|^{4} ≤ \frac{4sϵ^{8}}{m^{2}} ∑_{l=1}^{s} 𝔼h̄^{4}(X, ȳ_{l}).

Fixing x = X and taking into account (3.2) – (3.3) we can write

𝔼h̄^{4}(x, Ȳ_{l}) = 𝔼\{\frac{1}{\sqrt{2m}} ∑_{j=1}^{m} h̄(x, ˜ X̄_{j})\}^{4}
= \frac{1}{4m}𝔼h̄^{4}(x, ˜ X̄) + \frac{3(m − 1)}{4m}(𝔼h̄^{2}(x, ˜ X̄))^{2}
≤ \frac{2}{m}𝔼h̄^{4}(x, X̄) + 3(𝔼h̄^{2}(x, X̄))^{2}.

Here

𝔼h̄^{4}(X, X̄) ≤ 4^{3}𝔼Φ^{4}I(|Φ| ≤ m) + 3 · 4^{3}𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{m})

and

𝔼(𝔼X̄ h̄^{2}(X, X̄))^{2} ≤ 4^{4}𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{m}).

Since the random variable h̄(G, ȳ_{l}) is Gaussian then for 𝔼|ζ_{j}|^{4} we have the bounds

𝔼|ζ_{j}|^{4} ≤ \frac{4sϵ^{8}}{m^{2}} ∑_{l=1}^{s} 𝔼h̄^{4}(G, ȳ_{l})
= \frac{12sϵ^{8}}{m^{2}} ∑_{l=1}^{s}(𝔼h̄^{2}(X, ȳ_{l}))^{2}
≤ \frac{12sϵ^{8}}{m^{2}} ∑_{l=1}^{s} 𝔼h̄^{4}(X, y_{l}).

Therefore from (3.17) – (3.19) we see that

𝔼J(Ȳ) ≤ 3 · 4^{3}s^{2}λ_{m} + 3^{2}4^{3}5s^{2}γ_{m}.  (3.20)

A similar bound is valid for 𝔼J(G), that is,

𝔼J(G) ≤ 15 · 4^{3}s^{2}γ_{m}.  (3.21)

Combining (3.16), (3.20) and (3.21) we get the estimate (3.15). Further we shall show that in (3.12)

ℙ{|𝔸(ϵG, h̄) − 𝕀|_{∞} ≤ 2δ} ≥ ℙ{|𝔸(ϵG, h) − 𝕀|_{∞} ≤ δ} − 5(L_{m} + L_{n}).  (3.22)

Arguing as in (3.9) – (3.11) with G replacing Y we have

ℙ{|𝔸(ϵG, h̄) − 𝕀|_{∞} ≤ 2δ}  (3.23)
≥ ℙ\{|𝔸(ϵG, h) − 𝕀|_{∞} ≤ δ\} − \frac{1}{128}(𝔼ϕ^{2}(G, Ḡ) + 𝔼ϕ̄^{2}(G, Ḡ)).

Simple calculations show that

𝔼ϕ^{2}(G, Ḡ) = 𝔼θ^{2}(G, Ḡ) + 2𝔼(𝔼_{X}θ(X, G))^{2} + (𝔼θ(X, X̄))^{2}.  (3.24)

In (3.24), using (1.8)

𝔼θ^{2}(G, Ḡ) ≤ 12𝔼∥G∥^{2}I(∥G∥ > \sqrt{n}) + 4𝔼Φ^{2}(G, Ḡ)I(|Φ(G, Ḡ)| > n) (3.25)
≤ \frac{12}{n}𝔼∥G∥^{4} + \frac{4}{n}𝔼Φ^{4}(G, Ḡ)
= \frac{24}{n}𝔼(𝔼_{X}Φ^{2}(X, G))^{2}
= \frac{24}{n}𝔼(∑_{j=1}^{∞} q_{j}^{2}τ_{j}^{2})^{2} ≤ \frac{72}{n} ≤ 144(α_{n} + γ_{n}).

Similarly

𝔼θ^{2}(G, Ḡ) ≤ \frac{1}{n^{4}} (12𝔼∥G∥^{10} + 4𝔼Φ^{6}(G, Ḡ))  (3.26)
≤ \frac{612}{n^{4}} ≤ \frac{5000}{n^{2}}(α_{n}^{2} + γ_{n}^{2}).

Further in (3.24) we have

𝔼(𝔼_{X}θ(G, X))^{2}  (3.27)
≤ \frac{2}{n}𝔼(𝔼_{X}|Φ(G, X) |∥X∥ I(∥X∥ > \sqrt{n}))^{2}

+\frac{2}{n^{6}} 𝔼(𝔼_{X}Φ^{4}(G, X)I(∥X∥ > \sqrt{n}))^{2}

= \frac{2}{n}𝔼∥X∥ ∥X̄∥ I(∥X∥ > \sqrt{n})I(∥X̄∥ > \sqrt{n})𝔼_{G}|Φ(G, X)| |Φ(G, X̄)|

+\frac{2}{n^{6}} 𝔼I(∥X∥ ≤ \sqrt{n})I(∥X̄∥ ≤ \sqrt{n})𝔼_{G}Φ^{4}(G, X)Φ^{4}(G, X̄)

≤ \frac{2}{n}𝔼∥X∥ ∥X̄∥ I(∥X∥ > \sqrt{n})I(∥X̄∥ > \sqrt{n})(𝔼_{G}Φ^{2}(G, X))^{1/2}(𝔼_{G}Φ^{2}(G, X̄))^{1/2}

+\frac{2}{n^{6}} 𝔼I(∥X∥ ≤ \sqrt{n})I(∥X̄∥ ≤ \sqrt{n})(𝔼_{G}Φ^{8}(G, X))^{1/2}(𝔼_{G}Φ^{8}(G, X̄))^{1/2}

= \frac{2}{n}α_{n}^{2} + \frac{210}{n}γ_{n}^{2},

since the random variable Φ(G, x) is Gaussian. Combining (2.19) and (3.24) – (3.26) we obtain the inequalities

𝔼ϕ^{2}(G, Ḡ) ≤ 600(α_{n} + β_{n} + γ_{n}), 𝔼ϕ^{2}(G, Ḡ) ≤ \frac{6000}{n^{2}}(α_{n}^{2} + β_{n}^{2} + γ_{n}^{2}). (3.28)

The bound (3.22) follows from (3.23) and the first inequality (3.28). From (3.8), (3.12) and (3.22) we see that for all m ≥ 1

ℙ{|𝔸(ϵY, h) − 𝕀|_{∞} ≤ 8δ} ≥ ℙ{|𝔸(ϵG, h) − 𝕀|_{∞} ≤ δ} − 6L_{n} − 2^{12}s^{3}L_{m}. (3.29)

By the assumption (1.11), definition (3.6) and (3.29), if m ≥ m(s, q_{s}) then (3.7) holds. This proves Lemma 3.1.

4 Estimates of characteristic functions

In what follows we shall use the abridged notation U_{n}(Φ) or U(Φ; X_{1}, …, X_{n}) for Ustatistics of the type (1.1) to show the dependence on Φ or X_{1}, …, X_{n}. For example, using (2.14)

U_{n}(g) = U_{n}(h) + S + s,  (4.1)

where

S = \frac{n − 1}{n} ∑_{j=1}^{n}(𝔼X̄g(X_{j}, X̄) − 𝔼g(X, X̄)), s = \frac{1}{2}(n − 1)𝔼g(X, X̄)

and

g(x, y) = Φ(x, y)I(|Φ(x, y)| ≤ n)I(∥x∥ ≤ \sqrt{n})I(∥y∥ ≤ \sqrt{n}), x, y ∈ 𝔛.

Define the random variable

Φ_{∞}(G, G) = ∑_{j=1}^{∞} q_{j}(τ_{j}^{2} − 1).  (4.2)

Here the series converges in L^{2} and moreover 𝔼Φ_{∞}^{2}(G, G) = 2𝔼Φ^{2}(X, X̄). Substituting G =^{d} (G_{1} + ⋯ + G_{n})/\sqrt{n} and using (4.2) we have

Φ_{∞}(G, G) d = \frac{2}{n} ∑_{1≤i<j≤n} Φ(G_{i}, G_{j}) + \frac{1}{n} ∑_{j=1}^{n} Φ_{∞}(G_{j}, G_{j})

with Φ(G, Ḡ) = (ℚG, Ḡ). Then for the random variable U_{∞} = U_{∞}(Φ) defined by (1.4) we can write

U_{∞}(Φ) d = U(Φ; G_{1}, …, G_{n}) + \frac{1}{2n} ∑_{j=1}^{n} Φ_{∞}(G_{j}, G_{j}).  (4.3)

Note that the representation of U_{∞} equivalent to (4.3) can be given in the form of stochastic integrals (see, for example, Borovskikh(1996)).

According to (2.15) and (4.3)

U_{∞}(Φ) = U_{∞}(h) − U_{∞}(ϕ).  (4.4)

Now we shall estimate the difference of the characteristic functions

𝔼e{tU_{n}(g)} − 𝔼e{tU_{∞}(Φ)}, t ∈ ℝ,  (4.5)

where e{x} = exp{ix}, x ∈ ℝ. With the help of (4.1) and (4.4) write

𝔼e{tU_{n}(g)} = 𝔼e{tU_{n}(h)} + ϱ_{1}(t),
𝔼e{tU_{∞}(Φ)} = 𝔼e{tU_{∞}(h)} + ϱ_{2}(t),  (4.6)

where

ϱ_{1}(t) = 𝔼e{tU_{n}(h)}(e{t(S + s)} − 1),
ϱ_{2}(t) = 𝔼e{tU_{∞}(h)}(e{−tU_{∞}(ϕ)} − 1).

Further we shall use the abbreviations

f = th/n, f_{∞} = th_{∞}/(2n)  (4.7)

and the following bound κ(t) = κ_{1}(t) + κ_{2}(t) of characteristic functions, where

κ_{1}(t) = sup_{L}|𝔼e{ ∑_{1≤i<j≤k} f(X_{i}, X_{j}) + L(X_{1}, …, X_{k})}|,

κ_{2}(t) = sup_{L}|𝔼e{ ∑_{1≤i<j≤k} f(G_{i}, G_{j}) + L(G_{1}, …, G_{k})}|,

and the supremum is taken over all linear statistics L, of the form L(x_{1}, …, x_{k}) = l_{1}(x_{1}) + ⋯ + l_{k}(x_{k}) for some functions l_{1}, …, l_{k}, and k = [(n − 2)/20] is the integer part of (n − 2)/20.

Below we write A ≪ B if A ≤ cB and we shall use Taylor’s expansion for the function e{x} in the form

e{x} = ∑_{s=0}^{r−1}(ix)^{s}/s! + 𝔼(1 − v)^{r−1}e{vx}(ix)^{r}/(r − 1)! , x ∈ ℝ  (4.8)

where v is a random variable uniformly distributed on [0, 1] and independent of all other random variables.

Lemma 4.1. For t ∈ ℝ

|ϱ_{1}(t)| ≪ (|t| + t^{2})(α_{n} + β_{n} + γ_{n})κ(t).  (4.9)

Proof. Using the Taylor expansion (4.8) with r = 2

ϱ_{1}(t) = it 𝔼Se{tU_{n}(h)} + its𝔼e{tU_{n}(h)}  (4.10)
−t^{2}𝔼(1 − v)(S + s)^{2}e{tU_{n}(h) + vt(S + s)}.

Here by the i.i.d. assumption

𝔼Se{tU_{n}(h)}

= (n − 1)𝔼(𝔼X̄g(X_{1}, X̄) − 𝔼g(X, X̄))e\{\frac{t}{n} ∑_{j=2}^{n} h(X_{1}, X_{j}) + Ū(X_{2}, …, X_{n})\},

where

Ū(X_{2}, …, X_{n}) = ∑_{2≤i<j≤n} f(X_{i}, X_{j}).

Again by (4.8) with r = 1 we obtain

𝔼 Se\{tU_{n}(h)\} = \frac{n − 1}{n}it𝔼(𝔼X̄g(X_{1}, X̄) − 𝔼g(X, X̄))  (4.11)

×(∑_{j=2}^{n} h(X_{1}, X_{j}))e\{\frac{vt}{n} ∑_{j=2}^{n} h(X_{1}, X_{j}) + Ū(X_{2}, …, X_{n})\}.

Let

S_{1} = ∑_{2≤j≤n/2} h(X_{1}, X_{j}), S_{2} = ∑_{n/2<j≤n} h(X_{1}, X_{j}).

Then from (4.11) we get

|𝔼 Se{tU_{n}(h)}| ≤ |t|κ(t){𝔼|𝔼X̄g(X_{1}, X̄) − 𝔼g(X, X̄)|(|S_{1}| + |S_{2}|)}.  (4.12)

For fixed X_{1} = x we write taking into account the i.i.d. assumption, the degeneracy of h and Hölder’s inequality

𝔼(|S_{1}| + |S_{2}|) = 𝔼| ∑_{2≤j≤n/2} h(x, X_{j})| + 𝔼| ∑_{n/2<j≤n} h(x, X_{j})| ≤ 2\sqrt{n}(𝔼_{X}h^{2}(x, X))^{1/2}.

By virtue of definition of the function g(x, y) in Lemma 2.1

𝔼X̄g(x, X̄) = −𝔼X̄Φ(x, X̄)I(∥X̄∥ > \sqrt{n})I(∥x∥ ≤ \sqrt{n})  (4.13)
−𝔼X̄Φ(x, X̄)I(|Φ(x, X̄)| > n)I(∥X̄∥ ≤ \sqrt{n})I(∥x∥ ≤ \sqrt{n})

and

|𝔼X̄g(x, X̄)| ≤ \frac{1}{\sqrt{n}}𝔼X̄|Φ(x, X̄)|∥X̄∥I(∥X̄∥ > \sqrt{n})I(∥x∥ ≤ \sqrt{n}) (4.14)

+\frac{1}{n}𝔼X̄Φ^{2}(x, X̄)I(|Φ(x, X̄)| > n)I(∥x∥ ≤ \sqrt{n}),

|𝔼g(X, X̄)| ≤ \frac{1}{n}(α_{n} + β_{n}),  (4.15)

According to (2.14)

𝔼_{X}h^{2}(x, X) ≤ 8∥x∥^{2}I(∥x∥ ≤ \sqrt{n}) + 4𝔼_{X}(𝔼X̄g(X, X̄))^{2}  (4.16)
≤ 8(∥x∥^{2}I(∥x∥ ≤ \sqrt{n}) + α_{n} + β_{n}), x ∈ 𝔛.

Hence in (4.12)

𝔼|𝔼X̄g(X_{1}, X̄) − 𝔼g(X, X̄)|(|S_{1}| + |S_{2}|)  (4.17)
≪ 𝔼_{X}\{(∥X∥^{2} + α_{n} + β_{n})^{1/2}(𝔼X̄|Φ(X, X̄)| ∥X̄∥ I(∥X̄∥ > \sqrt{n})

+\frac{1}{\sqrt{n}}𝔼X̄Φ^{2}(X, X̄)I(|Φ(X, X̄)| > n)I(∥X∥ ≤ \sqrt{n}) + α_{n} + β_{n})\}.

For fixed X̄ = x we have by Hölder’s inequality

𝔼(∥X∥^{2} + α_{n} + β_{n})^{1/2}|Φ(X, x)| ≤ ∥x∥,

that is in (4.17)

𝔼_{X}\{(∥X∥^{2} + α_{n} + β_{n})^{1/2}𝔼X̄|Φ(X, X̄)| ∥X̄∥ I(∥X̄∥ > \sqrt{n}) ≤ 2α_{n}.

Using the inequality ab ≤ a^{2} + b^{2} we see that in (4.17)

\frac{1}{\sqrt{n}}𝔼_{X}(∥X∥^{2} + α_{n} + β_{n})^{1/2}𝔼X̄Φ^{2}(X, X̄)I(|Φ(X, X̄)| > n)I(∥X∥ ≤ \sqrt{n})

= 𝔼\{\frac{1}{\sqrt{n}}(∥X∥^{2} + α_{n} + β_{n})^{1/2}|Φ(X, X̄)|I(∥X∥ ≤ \sqrt{n})\}
×{|Φ(X, X̄)| I(|Φ(X, X̄)| > n)}
≤ \frac{1}{n}(𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{n}) + α_{n} + β_{n}) + β_{n}.

Thus in (4.12)

|𝔼 Se{tU_{n}(h)}| ≪ |t|κ(t)(α_{n} + β_{n} + γ_{n}).  (4.18)

The second expectation in (4.10) is bounded using (4.14), (4.15) and κ(t). Consider the third expectation in (4.10). Split S + s = s_{1} + s_{2} + s_{3}, where

s_{1} = s + ∑_{1≤j≤n/3} φ(X_{j}), s_{2} = ∑_{n/3<j≤2n/3} φ(X_{j}),

and s_{3} = S + s − s_{1} − s_{2}, and φ(x) = (𝔼X̄g(x, X̄) − 𝔼g(X, X̄))(n − 1)/n. Then in (4.10)

|𝔼^{v}(S + s)^{2}e{tU_{n}(h) + vt(S + s)}| ≤ ∑_{1≤j,k≤3}|𝔼^{v}s_{j}s_{k}e{tU_{n}(h) + vt(S + s)}|,

where 𝔼^{v} denotes the conditional expectation given v. For example, consider s_{1}s_{2}. Conditioning on s_{1} and s_{2}, using the i.i.d. assumption and the definition of κ(t) we obtain

|𝔼^{v}s_{1}s_{2}e{tU_{n}(h) + vt(S + s)}|  (4.19)
≤ 𝔼^{v}|s_{1}| |s_{2}| |𝔼_{s_{3}}e{tU_{n}(h) + vts_{3}}|
≤ (𝔼s_{1}^{2} + 𝔼s_{2}^{2})κ(t) ≪ (n𝔼_{X}(𝔼X̄g(X, X̄))^{2} + n^{2}(𝔼g(X, X̄))^{2})κ(t).

Here by (4.13)

𝔼_{X}(𝔼X̄g(X, X̄))^{2}  (4.20)
≤ 2𝔼_{X}(𝔼X̄Φ(X, X̄)I(∥X̄∥ > \sqrt{n}))^{2}

+2𝔼_{X}(𝔼X̄Φ(X, X̄)I(|Φ(X, X̄)| > n)I(∥X̄∥ ≤ \sqrt{n})I(∥X∥ ≤ \sqrt{n}))^{2}

≤ \frac{2}{n}\{𝔼_{X}(𝔼X̄|Φ(X, X̄)| ∥X̄∥ I(∥X̄∥ > \sqrt{n}))^{2}

+\frac{1}{n}𝔼_{X}(𝔼X̄Φ^{2}(X, X̄))^{2}I(∥X∥ ≤ \sqrt{n})\} ≪ \frac{1}{n}(α_{n} + γ_{n}),

where

𝔼_{X}(𝔼X̄|Φ(X, X̄)| ∥X̄∥ I(∥X̄∥ > \sqrt{n}))^{2}

= 𝔼X̄_{1}𝔼X̄_{2}∥X̄_{1}∥ ∥X̄_{2}∥I(∥X̄_{1}∥ > \sqrt{n})I(∥X̄_{2}∥ > \sqrt{n})𝔼_{X}|Φ(X, X̄_{1})| |Φ(X, X̄_{2})|
≤ α_{n}^{2} ≤ α_{n}.

Combining (4.10), (4.18) – (4.20) we obtain (4.9).

Lemma 4.2. For t ∈ ℝ

|ϱ_{2}(t)| ≪ (α_{n} + β_{n} + γ_{n})((|t| + t^{2})κ(t) + |t|n^{−3}).  (4.21)

Proof. According to (2.15) and (4.3) we can write the representation

U_{∞}(ϕ) = U(θ; G_{1}, …, G_{n}) + S(θ) + S_{0},

where

S(θ) = ∑_{j=1}^{n} θ_{j}, θ_{j} = −\frac{n − 1}{n}𝔼_{X}θ(G_{j}, X) + \frac{1}{2n}ϕ_{∞}(G_{j}, G_{j}),

S_{0} = \frac{1}{2}(n − 1)𝔼θ(X, X̄).

Applying (4.8) three times and taking account of this representation we obtain

ϱ_{2}(t) = −it𝔼S(θ)e{tU_{∞}(h)} − t^{2}ϱ_{21}(t) − itϱ_{22}(t) − itϱ_{23}(t),  (4.22)

where

ϱ_{21}(t) = 𝔼(1 − v)S^{2}(θ)e{tU_{∞}(h) − vtS(θ)},
ϱ_{22}(t) = 𝔼U(θ; G_{1}, …, G_{n})e{tU_{∞}(h) − tS(θ) − tS_{0} − vtU(θ; G_{1}, …, G_{n})}, and
ϱ_{23}(t) = S_{0}𝔼e{tU_{∞}(h) − tS(θ) − vtS_{0}}.

First estimate ϱ_{21}(t). Split the sum S(θ) into the three parts

S(θ) = s_{1}(θ) + s_{2}(θ) + s_{3}(θ),

where

s_{1}(θ) = ∑_{1≤j≤n/3} θ_{j}, s_{2}(θ) = ∑_{n/3<j≤2n/3} θ_{j}, s_{3}(θ) = S(θ) − s_{1}(θ) − s_{2}(θ).

Just as in the proof of (4.19) we have

|ϱ_{21}(t)| ≪ ∑_{j=1}^{3} 𝔼s_{j}^{2}(θ)κ(t),  (4.23)

where

𝔼s_{j}^{2}(θ) ≪ n𝔼(𝔼_{X}θ(G, X))^{2} + \frac{1}{n}𝔼ϕ_{∞}^{2}(G, G).

Here according to (1.3), (2.5), (2.15) and (4.2)

𝔼ϕ_{∞}^{2}(G, G) = 𝔼(Φ_{∞}(G, G) − h_{∞}(G, G))^{2}
= 𝔼(∑_{j=1}^{∞}(q_{j} − q̄_{j})(τ_{j}^{2} − 1))^{2}
= 2 ∑_{j=1}^{∞}(q_{j} − q̄_{j})^{2}
= 2𝔼(∑_{j=1}^{∞}(q_{j} − q̄_{j})τ_{j}τ̄_{j})^{2}
= 2𝔼(Φ(G, Ḡ) − h(G, Ḡ))^{2} = 2𝔼ϕ^{2}(G, Ḡ).

With the help of the second inequality (3.28) we obtain

𝔼ϕ_{∞}^{2}(G, G) ≤ \frac{12000}{n^{2}}(α_{n}^{2} + β_{n}^{2} + γ_{n}^{2}).  (4.24)

Thus by (3.27) and (4.24) in (4.23)

|ϱ_{21}(t)| ≪ (α_{n} + β_{n} + γ_{n})κ(t).  (4.25)

To estimate ϱ_{22}(t) we write

|ϱ_{22}(t)| ≤ 𝔼|U(θ; G_{1}, …, G_{n})| ≤ n𝔼|θ(G, Ḡ)|  (4.26)
≤ 3n 𝔼∥G∥ I(∥G∥ > \sqrt{n}) + n 𝔼|Φ(G, Ḡ)| I(|Φ(G, Ḡ)| > \sqrt{n})
≤ \frac{3}{n^{4}} (𝔼∥G∥^{12} + 𝔼Φ^{6}(G, Ḡ)) ≪ \frac{1}{n^{4}} .

Further we have taking into account (2.19)

|ϱ_{23}(t)| ≤ n |𝔼θ(X, X̄)|κ(t) ≤ (α_{n} + β_{n})κ(t).  (4.27)

Next we bound the expectation in (4.22). By the i.i.d. assumption and (4.8) we have

𝔼S(θ)e{tU_{∞}(h)}  (4.28)
= −(n − 1)𝔼 𝔼_{X}θ(G, X)e\{\frac{t}{n} ∑_{j=2}^{n} h(G, G_{j}) + Ū(G_{2}, …, G_{n})\}

+\frac{1}{2}𝔼 ϕ_{∞}(G, G)e\{\frac{t}{n} ∑_{j=2}^{n} h(G, G_{j}) + Ū(G_{2}, …, G_{n})\}

= −(n − 1)𝔼θ(G, X) 𝔼e\{Ū(G_{2}, …, G_{n})\} − \frac{n − 1}{n}itϱ_{3}(t) + \frac{it}{2n}ϱ_{4}(t),

where by (2.17)

𝔼θ(G, X) = 𝔼Φ(G, X)I(∥G∥ > \sqrt{n})I(∥X∥ > \sqrt{n})
−𝔼Φ(G, X)I(|Φ(G, X)| > n)I(∥G∥ ≤ \sqrt{n})I(∥X∥ ≤ \sqrt{n})

and

ϱ_{3}(t) = 𝔼 𝔼_{X}θ(G, X) ∑_{j=2}^{n} h(G, G_{j})e\{\frac{vt}{n} ∑_{j=2}^{n} h(G, G_{j}) + Ū(G_{2}, …, G_{n})\},
ϱ_{4}(t) = 𝔼 ϕ_{∞}(G, G) ∑_{j=2}^{n} h(G, G_{j})e\{\frac{vt}{n} ∑_{j=2}^{n} h(G, G_{j}) + Ū(G_{2}, …, G_{n})\}.

Just as in (4.11) – (4.12) since 𝔼Φ^{2}(X, X̄) = 1, 𝔼h^{2}(X, X̄) ≤ 16 and (3.27) we find

|ϱ_{3}(t)| ≪ \sqrt{n}(𝔼(𝔼_{X}θ(G, X))^{2})^{1/2}κ(t) ≪ (α_{n} + γ_{n})κ(t)  (4.29)

Similarly for ϱ_{4}(t) with using (4.24) we find

|ϱ_{4}(t)| ≪ \sqrt{n}(𝔼ϕ_{∞}^{2}(G, G))^{1/2}κ(t) ≪ (α_{n} + β_{n} + γ_{n})κ(t).  (4.30)

From (4.22) – (4.30) we obtain (4.21).

In what follows we shall estimate the difference of the characteristic functions

𝔼e{tU_{n}(h)} − 𝔼{tU_{∞}(h)}, t ∈ ℝ

when the kernel is degenerate. Just as in Bentkus and Götze (1999) we shall apply the Bergström type identity

𝔼e{tU_{n}(h)} − 𝔼{tU_{∞}(h)}  (4.31)
= 𝔼e{ℜ_{[1,n]}tU_{n}(h)} − 𝔼e{tU_{∞}(h)} + n(𝔼e{ℜ_{[2,n]}tU_{n}(h)} − 𝔼e{ℜ_{[1,n]}tU_{n}(h)})

+ ∑_{k=2}^{n}(k − 1)(𝔼e{ℜ_{[3,k]}tU_{n}(h)} − 𝔼e{ℜ_{[2,k]}tU_{n}(h)}

−𝔼e{ℜ_{[1,k]\{2}}tU_{n}(h)} + 𝔼e{ℜ_{[1,k]}tU_{n}(h)}),

where the operator ℜ_{A} replaces the random variables X_{j}, j ∈ A, in U(h; X_{1}, …, X_{n}) by G_{j}, j ∈ A. As a result of these replacements in (4.31)

ℜ_{[1,k]}tU_{n}(h) = f(G_{1}, S_{k}) + f(G_{2}, S_{k}) + f(G_{1}, G_{2}) + V_{k},
ℜ_{[2,k]}tU_{n}(h) = f(X_{1}, S_{k}) + f(G_{2}, S_{k}) + f(X_{1}, G_{2}) + V_{k},
ℜ_{[3,k]}tU_{n}(h) = f(X_{1}, S_{k}) + f(X_{2}, S_{k}) + f(X_{1}, X_{2}) + V_{k},
ℜ_{[1,k]\{2}}tU_{n}(h) = f(G_{1}, S_{k}) + f(X_{2}, S_{k}) + f(G_{1}, X_{2}) + V_{k},

where

S_{k} = ∑_{j=k+1}^{n} X_{j} + ∑_{j=3}^{k} G_{j}, and

V_{k} = ∑_{i=3}^{k−1} ∑_{j=i+1}^{k} f(G_{i}, G_{j}) + ∑_{i=3}^{k} ∑_{j=k+1}^{n} f(G_{i}, X_{j}) + ∑_{i=k+1}^{n−1} ∑_{j=k+1}^{n} f(X_{i}, X_{j}).

Following Bentkus and Götze (1999) we shall now estimate the difference of the expectations on the right in (4.31). Under the condition 𝔼Φ^{2}(X, X̄) = 1 the bounds depend on the moments of the truncated kernel h.

Lemma 4.3. For t ∈ ℝ

|𝔼e\{ℜ_{[1,n]}tU_{n}(h)\} − 𝔼e\{tU_{∞}(h)\}| ≪ \frac{1}{n}(t^{2} + |t|^{3})κ(t),  (4.32)

and

n|𝔼e{ℜ_{[2,n]}tU_{n}(h)} − 𝔼e{ℜ_{[1,n]}tU_{n}(h)}| ≪ γ_{n}t^{4}κ(t).  (4.33)

Proof. According to (4.3), (4.4)

tU_{∞}(h) = tℜ_{[1,n]}U_{n}(h) + 𝔘, where 𝔘 = ∑_{j=1}^{n} f_{∞}(G_{j}, G_{j}).

Using (4.8) with r = 2 we write

𝔼e{ℜ_{[1,n]}tU_{n}(h)} − 𝔼e{tU_{∞}(h)}
= i𝔼 𝔘e{ℜ_{[1,n]}tU_{n}(h)} − 𝔼𝔘^{2}(1 − v)e{ℜ_{[1,n]}tU_{n}(h) + v𝔘}.  (4.34)

By the i.i.d. assumption

𝔼 𝔘e{ℜ_{[1,n]}tU_{n}(h)} = n𝔼f_{∞}(G, G)e{S + Ū(G_{2}, …, G_{n})},

where

S = f_{∞}(G, G_{2}) + ⋯ + f_{∞}(G, G_{n}).

Further taking into account 𝔼f_{∞}(G, G) = 0 and 𝔼_{G}f_{∞}(G, G)f_{∞}(G, G_{j}) = 0, 2 ≤ j ≤ n, we get by (4.8)

|𝔼 𝔘e{ℜ_{[1,n]}tU_{n}(h)}| ≤ n𝔼|𝔼^{v}f_{∞}(G, G)S^{2}e{vS + Ū(G_{2}, …, G_{n})}|.

To estimate the expectation on the right we repeat the reasoning used for the estimation of the expectation in (4.10). As a result we find

|𝔼 𝔘e\{ℜ_{[1,n]}tU_{n}(h)\}| ≪ \frac{1}{n}|t|^{3}κ(t).  (4.35)

Similarly spliting the sum 𝔘 in (4.34) into a sum of three approximately equal parts we obtain the bound

|𝔼 𝔘^{2}(1 − v)e\{ℜ_{[1,n]}tU_{n}(h) + v𝔘\}| ≪ \frac{1}{n}|t|^{2}κ(t).  (4.36)

The bound (4.32) follows from (4.34) – (4.36). The bound (4.33) follows from the inequality (8.61) in Bentkus and Götze (1999).

n|𝔼e\{ℜ_{[2,n]}tU_{n}(h)\} − 𝔼e\{ℜ_{[1,n]}tU_{n}(h)\}| ≪ \frac{1}{n}𝔼(𝔼_{X}h^{2}(X, X̄))^{2}t^{4}κ(t) ≪ γ_{n}t^{4}κ(t),

since β̄_{4} = 0 and F̂_{1}(t) = 0 due to the symmetry of G_{j} and the linearity of h(x, y). This proves Lemma 4.3.

It remains to estimate the sum over 2 ≤ k ≤ n on the right in (4.31). Denote

J_{1}(k) = 𝔼e{f(X_{1}, S_{k}) + f(X_{2}, S_{k}) + f(X_{1}, X_{2}) + V_{k}},
J_{2}(k) = 𝔼e{f(G_{1}, S_{k}) + f(X_{2}, S_{k}) + f(G_{1}, X_{2}) + V_{k}},
J_{3}(k) = 𝔼e{f(X_{1}, S_{k}) + f(G_{2}, S_{k}) + f(X_{1}, G_{2}) + V_{k}},
J_{4}(k) = 𝔼e{f(G_{1}, S_{k}) + f(G_{2}, S_{k}) + f(G_{1}, G_{2}) + V_{k}}.

Note that by the i.i.d. assumption J_{2}(k) = J_{3}(k).

Lemma 4.4. For t ∈ ℝ

∑_{k=2}^{n} k|J_{1}(k) − J_{2}(k) − J_{3}(k) + J_{4}(k)| ≪ (|t| + t^{6})κ(t)(L_{n} + μ_{n}),  (4.37)

where μ_{n} = n^{−1}|𝔼Φ^{3}(X, X̄)I(|Φ(X, X̄)| ≤ n)|.

Proof. To prove (4.37) it suffices to verify that

|J_{1}(k) − J_{2}(k) − J_{3}(k) + J_{4}(k)| ≪ \frac{1}{n^{2}} (L_{n} + μ_{n})(|t| + t^{6})κ(t).  (4.38)

To simplify the notation we shall provide the arguments for k = 2. The case k > 2 is similar since here in S_{k} and V_{k} the random vectors X_{3}, …, X_{k} are replaced by the Gaussian random vectors G_{3}, …, G_{k}. For the case k = 2 introduce the notation

x̂ = ∑_{1≤j≤n−2} f(x, X_{j}), x ∈ 𝔛

and let V = V_{2}, J_{s} = J_{s}(2), 1 ≤ s ≤ 4. Further replacing X_{1}, X_{2}, G_{1}, G_{2}, X_{3}, …, X_{n}, G_{3}, …, G_{n} by X, X̄, G, Ḡ, X_{1}, …, X_{n−2}, G_{1}, …, G_{n−2} respectively, we can rewrite J_{s}, 1 ≤ s ≤ 4, as in Bentkus and Götze (1999), in the form

J_{1} = 𝔼e{V + X̂ + ˆ X̄ + f(X, X̄)},
J_{2} = 𝔼e{V + Ĝ + ˆ X̄ + f(G, X̄)},
J_{3} = 𝔼e{V + X̂ + ˆ Ḡ + f(X, Ḡ)},
J_{4} = 𝔼e{V + Ĝ + ˆ Ḡ + f(G, Ḡ)},

where J_{2} = J_{3}. Expanding via Taylor series in powers of f(·, ·) up to terms of order O(f^{4}) and conditioning, we get

|J_{s} − J_{s0} − J_{s1} − \frac{1}{2}J_{s2} − \frac{1}{6}J_{s3}| ≪ 𝔼f^{4}(X, X̄)κ(t), 1 ≤ s ≤ 4,  (4.39)

where we use the fact that

𝔼f^{4}(G, Ḡ) ≤ 3𝔼f^{4}(X, G) ≤ 9𝔼f^{4}(X, X̄).

Denoting I_{r} = J_{1r} − J_{2r} − J_{3r} + J_{4r}, 0 ≤ r ≤ 3, and using (4.39), we obtain

|J_{1} − J_{2} − J_{3} + J_{4}| ≪ |I_{0}| + |I_{1}| + |I_{2}| + |I_{3}| + 𝔼f^{4}(X, X̄)κ(t).

Hence, instead of (4.38) it suffices to verify that

|I_{r}| ≪ (|t| + t^{6})κ(t)\frac{1}{n^{2}} (L_{n} + μ_{n}), for 0 ≤ r ≤ 3.  (4.40)

We shall apply the following Rosenthal type inequality: for independent random variables ξ_{1}, …, ξ_{n} in ℝ with mean zero

𝔼| ∑_{j=1}^{n} ξ_{j}|^{q} ≤ c(q)(∑_{j=1}^{n} 𝔼|ξ_{j}|^{q} + (∑_{j=1}^{n} 𝔼ξ_{j}^{2})^{q/2}), 2 ≤ q < ∞.  (4.41)

Using the definition of I_{r} we write, for 0 ≤ r ≤ 3,

|I_{r}| ≪ |𝔼e{V }(φ_{r}(X, X̄) − φ_{r}(G, X̄) − φ_{r}(X, Ḡ) + φ_{r}(G, Ḡ))|,  (4.42)

where

φ_{r}(x, y) = f^{r}(x, y)e{x̂ + ŷ}, x, y ∈ 𝔛.

Verification of (4.40) for r = 0. In (4.42) with the help of (4.8) and (1.15) we have

𝔼e{V }(φ_{0}(X, X̄) − φ_{0}(G, X̄) − φ_{0}(X, Ḡ) + φ_{0}(G, Ḡ))
= 𝔼e{V }(𝔼_{X}e{X̂} − 𝔼_{G}e{Ĝ})^{2}
= −\frac{1}{4}𝔼e\{V \}[𝔼_{v}(1 − v)^{2}(𝔼_{X} X̂^{3}e\{vX̂\} − 𝔼_{G}Ĝ^{3}e\{vĜ\}]^{2}
= −\frac{1}{4}𝔼(1 − v)^{2}(1 − v̄)^{2}[𝔼_{1}(X̂ ˆX̄)^{3}e\{V + vX̂ + v̄ ˆX̄\}
+2𝔼_{1}(X̂Ĝ)^{3}e{V + vX̂ + v̄Ĝ} − 𝔼_{1}(Ĝ ˆḠ)^{3}e{V + vĜ + v̄ˆḠ}],

where 𝔼_{1} denotes the expectation over X_{1}, …, X_{n−2}. From here and (4.42) we obtain for r = 0

|I_{0}| ≪ 𝔼|𝔼_{1}(X̂ ˆX̄)^{3}e{V + vX̂ + v̄ ˆX̄}|  (4.43)
+𝔼|𝔼_{1}(X̂Ĝ)^{3}e{V + vX̂ + v̄Ĝ}| + 𝔼|𝔼_{1}(Ĝ ˆḠ)^{3}e{V + vĜ + v̄ˆḠ}|.

We will now show that in (4.43)

𝔼|X̂ ˆX̄|^{3} ≪ t^{6} \frac{1}{n^{2}} L_{n}.  (4.44)

Similar inequalities hold for the expectations 𝔼|X̂Ĝ|^{3} and 𝔼|Ĝ ˆḠ|^{3}. To prove (4.44) we write

𝔼|X̂ ˆX̄|^{3} = t^{6} \frac{1}{n^{6}} 𝔼|(∑_{i=1}^{n−2} h(X, X_{i}))(∑_{j=1}^{n−2} h(X̄, X_{j}))|^{3}.  (4.45)

Here

(∑_{i=1}^{n−2} h(X, X_{i}))(∑_{j=1}^{n−2} h(X̄, X_{j})) = ∑_{j=1}^{n−2}(h(X, X_{j})h(X̄, X_{j}) − 𝔼_{Z}h(X, Z)h(X̄, Z))

+ ∑_{1≤i≠j≤n−2} h(X, X_{i})h(X̄, X_{j}) + (n − 2)𝔼_{Z}h(X, Z)h(X̄, Z).  (4.46)

Due to (4.46) and since |a + b + c|^{3} ≤ 9(|a|^{3} + |b|^{3} + |c|^{3}) we have

𝔼|(∑_{i=1}^{n−2} h(X, X_{i}))(∑_{j=1}^{n−2} h(X̄, X_{j}))|^{3}  (4.47)

≪ 𝔼|∑_{j=1}^{n−2}(h(X, X_{j})h(X̄, X_{j}) − 𝔼_{Z}h(X, Z)h(X̄, Z))|^{3}

+𝔼| ∑_{1≤i≠j≤n−2} h(X, X_{i})h(X̄, X_{j})|^{3} + n^{3}𝔼|𝔼_{Z}h(X, Z)h(X̄, Z)|^{3}.

By (4.41) and conditioning

𝔼|∑_{j=1}^{n−2}(h(X, X_{j})h(X̄, X_{j}) − 𝔼_{Z}h(X, Z)h(X̄, Z))|^{3}

≪ n𝔼|h(X, Z)h(X̄, Z)|^{3} + n^{3/2}𝔼(𝔼_{Z}h^{2}(X, Z)h^{2}(X̄, Z))^{3/2}.

Here by (2.14) and (4.16)

𝔼|h(X, Z)h(X̄, Z)|^{3} ≪ n^{2}𝔼_{Z}(𝔼_{X}h^{2}(X, Z))^{2}
≪ n^{2}𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{n}) + n^{2}(α_{n} + β_{n})^{2}

and

𝔼(𝔼_{Z}h^{2}(X, Z)h^{2}(X̄, Z))^{3/2} ≪ n𝔼(𝔼_{Z}h^{2}(X, Z)h^{2}(X̄, Z))(𝔼_{Z}h^{2}(X, Z))^{1/2}
≪ n^{3/2}𝔼_{Z}(𝔼_{X}h^{2}(X, Z))^{2}
≪ n^{3/2}(𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{n}) + α_{n} + β_{n}).

Further in (4.47) by the decoupling inequality for U-statistics (see, for example, de la Peña (1992))

𝔼| ∑_{1≤i≠j≤n−2} h(X, X_{i})h(X̄, X_{j})|^{3} ≪ 𝔼| ∑_{1≤i≠j≤n−2} h(X, X_{i})h(X̄, X_{j}^{1})|^{3},  (4.48)

where X_{j}^{1} is a independent copy of X_{j}. Hence in (4.48)

∑_{1≤i≠j≤n−2} h(X, X_{i})h(X̄, X_{j}^{1}) = ∑_{i=1}^{n−2} ξ_{i},  (4.49)

where

ξ_{i} = h(X, X_{i}) ∑_{{j=1 \atop j≠i}}^{n−2} h(X̄, X_{j}).

Again by (4.41) and conditioning we obtain

𝔼| ∑_{1≤i≠j≤n−2} h(X, X_{i})h(X̄, X_{j}^{1})|^{3} ≪ ∑_{i=1}^{n−2} 𝔼|ξ_{i}|^{3} + 𝔼(∑_{i=1}^{n−2} 𝔼_{X_{i}}ξ_{i}^{2})^{3/2},  (4.50)

where

∑_{i=1}^{n−2} 𝔼|ξ_{i}|^{3} = (n − 2)𝔼|h(X, Z)|^{3}𝔼| ∑_{j=1}^{n−3} h(X̄, X_{j}^{1})|^{3} and

𝔼(∑_{i=1}^{n−2} 𝔼_{X_{i}}ξ_{i}^{2})^{3/2} = (n − 2)^{3/2}𝔼(𝔼_{Z}h^{2}(X, Z))^{3/2}𝔼∥ ∑_{j=1}^{n−3} h(X̄, X_{j})∥^{3}.

Once more by (4.41) and conditioning

𝔼| ∑_{j=1}^{n−3} h(X̄, X_{j}^{1})|^{3} ≪ (n − 3)𝔼|h(X, X̄)|^{3} + (n − 3)^{3/2}𝔼(𝔼_{Z}h^{2}(X̄, Z))^{3/2}. (4.51)

From (4.48) – (4.51) we obtain

𝔼| ∑_{1≤i≠j≤n−2} h(X, X_{i})h(X̄, X_{j})|^{3}  (4.52)
≪ n^{2}(𝔼|h(X, X̄)|^{3})^{2} + n^{5/2}𝔼|h(X, X̄)|^{3}𝔼(𝔼_{X}h^{2}(X, X̄))^{3/2}
+n^{3}(𝔼(𝔼_{X}h^{2}(X, X̄))^{3/2})^{2}.

In (4.52) due to (2.14), (4.16) and ab ≤ a^{2} + b^{2}

(𝔼|h(X, X̄)|^{3})^{2} ≤ 𝔼h^{2}(X, X̄)𝔼h^{4}(X, X̄) ≪ n^{2}λ_{n},  (4.53)

and

\frac{𝔼|h(X, X̄)|^{3}}{n} \frac{𝔼(𝔼_{X}h^{2}(X, X̄))^{3/2}}{\sqrt{n}} ≤ \frac{(𝔼|h(X, X̄)|^{3})^{2}}{n^{2}} + \frac{(𝔼(𝔼_{X}h^{2}(X, X̄))^{3/2})^{2}}{n}
≪ λ_{n} + γ_{n},

since

(𝔼(𝔼_{X}h^{2}(X, X̄))^{3/2})^{2} ≪ (𝔼∥X∥^{3} I(∥X∥ ≤ \sqrt{n}))^{2} + α_{n} + β_{n}
≪ 𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{n}) + α_{n} + β_{n}.

Further in (4.47) by Hölder inequality

𝔼|𝔼_{Z}h(X, Z)h(X̄, Z)|^{3} ≤ (𝔼(𝔼_{Z}h^{2}(X, Z))^{3/2})^{2}.  (4.54)

From (4.45) – (4.54) we obtain (4.44).

Finally, to get (4.40) for r = 0 from (4.43), it sufficies to repeat the arguments leading to (4.19): partition the sums of the type X̂ in (4.43) into a sum of independent parts with approximately equal numbers of summands; condition on the part which is absent in the product to obtain the bound κ(t); then estimate the expectations by the analogy with (4.44). More specifically, partition [1, n−2] = N_{1}∪⋯∪N_{l} into an union of l disjoint subsets with approximately equal cardinalities. Further we write

X̂ = ∑_{i=1}^{l} a_{i}, a_{i} = ∑_{j∈N_{i}} f(X, X_{j}), and

ˆX̄ = ∑_{i=1}^{l} b_{i}, b_{i} = ∑_{j∈N_{i}} f(X̄, X_{j}).

It is well known that for l ≥ r

(x_{1} + … + x_{l})^{r} = ∑_{k=1}^{r} ∑_{1≤i_{1},<···<i_{k}≤l} ∑_{{ν_{1}+···+ν_{k}=r \atop ν_{j} ≥1}} \frac{r!}{ν_{1}! ⋯ ν_{k}!}x_{i_{1}}^{ν_{1}} ⋯ x_{i_{k}}^{ν_{k}}, x_{i} ∈ ℝ.

Apply this formula for x_{i} = a_{i} and x_{i} = b_{i} provided 2r < l < 20. By this formula the product

(a_{1} + ⋯ + a_{l})^{r}(b_{1} + ⋯ + b_{l})^{r}

is decomposed into a linear combination of terms of the form

a_{i_{1}}^{ν_{1}} ⋯ a_{i_{k}}^{ν_{k}} · b_{j_{1}}^{μ_{1}} ⋯ b_{j_{s}}^{μ_{s}}  (4.55)

with

ν_{1} + ⋯ + ν_{k} = μ_{1} + ⋯ + μ_{s} = r, ν_{i} ≥ 1, μ_{i} ≥ 1,

and

1 ≤ i_{1} < ⋯ < i_{k} ≤ l, 1 ≤ j_{1} < ⋯ < j_{s} ≤ l, 1 ≤ k, s ≤ r.

Since l > 2r then in the product (4.55) at least one pair (a_{q}, b_{q}) is absent. By conditioning on this pair (a_{q}, b_{q}) we obtain for 2r < l < 20

|𝔼_{1}(X̂ ˆX̄)^{r}e{V + vX̂ + v̄ ˆX̄}| ≪ (∑_{i=1}^{l} 𝔼_{1}|a_{i}b_{i}|^{r} + ∑_{1≤i≠j≤l} 𝔼_{1}|a_{i}|^{r}𝔼_{1}|b_{j}|^{r})κ(t). (4.56)

Applying (4.56) with r = 3 and l = 7

𝔼_{1}|a_{i}|^{3} ≤ 𝔼_{1}|X̂|^{3} = \frac{t^{3}}{n^{3}} 𝔼_{1}| ∑_{j=1}^{n−2} h(X, X_{j})|^{3}

≪ \frac{t^{3}}{n^{3}} (n 𝔼_{Z}|h(X, Z)|^{3} + n^{3/2}(𝔼_{Z}h^{2}(X, Z))^{3/2}).

Similar estimate is valid for 𝔼_{1}|b_{j}|^{3}. Hence for 1 ≤ i ≠ j ≤ l

𝔼|a_{i}|^{3}𝔼|b_{j}|^{3} ≪ \frac{t^{6}}{n^{2}} \{\frac{1}{n^{2}} (𝔼|h(X, Z)|^{3})^{2} + \frac{1}{n}(𝔼(𝔼_{Z}h^{2}(X, Z))^{3/2})^{2}\}
≪ \frac{t^{6}}{n^{2}} L_{n}.

The expectation 𝔼|a_{i}b_{i}|^{3} is estimated with the help of (4.44). As a result we find from (4.56) that in (4.43)

𝔼|𝔼_{1}(X̂ ˆX̄)^{3}e\{V + vX̂ + v̄ ˆX̄\}| ≪ t^{6}κ(t)\frac{1}{n^{2}} L_{n}.

This estimate is also valid for other similar expectations in (4.43). That is, we obtain (4.40) for r = 0.

Verification of (4.40) for r = 1. Denoting

α(x) = e{x} − ix − 1 = −x^{2}𝔼(1 − v)e{vx}, x ∈ ℝ

and using (1.15) we write in (4.42)

𝔼e{V }(φ_{1}(X, X̄) − φ_{1}(G, X̄) − φ_{1}(X, Ḡ) + φ_{1}(G, Ḡ))  (4.57)
= 𝔼e{V }[f(X, X̄)(e{X̂} − 1)(e{ ˆX̄} − 1) − f(G, X̄)(e{Ĝ} − 1)(e{ ˆX̄} − 1)
−f(X, Ḡ)(e{X̂} − 1)(e{ ˆḠ} − 1) + f(G, Ḡ)(e{Ĝ} − 1)(e{ ˆḠ} − 1)]
= 𝔼e{V }[f(X, X̄)(α(X̂) + iX̂)(α( ˆX̄) + i ˆX̄) − f(G, X̄)(α(Ĝ) + iĜ)(α( ˆX̄) + i ˆX̄)
−f(X, Ḡ)(α(X̂) + iX̂)(α( ˆḠ) + i ˆḠ) + f(G, Ḡ)(α(Ĝ) + iĜ)(α( ˆḠ) + i ˆḠ)]
= 𝔼e{V }[f(X, X̄)(α(X̂)α( ˆX̄) + iα( ˆX̄)X̂ + iα(X̂) ˆX̄ − X̂ ˆX̄)
−f(G, X̄)(α(Ĝ)α( ˆX̄) + iα( ˆX̄)Ĝ + iα(Ĝ) ˆX̄ − Ĝ ˆX̄)
−f(X, Ḡ)(α(X̂)α( ˆḠ) + iα( ˆḠ)X̂ + iα(X̂) ˆḠ − X̂ ˆḠ)
+f(G, Ḡ)(α(Ĝ)α( ˆḠ) + iα( ˆḠ)Ĝ + iα(Ĝ) ˆḠ − Ĝ ˆḠ)]
= 𝔼e{V }[f(X, X̄)α(X̂)α( ˆX̄) − f(G, X̄)α(Ĝ)α( ˆX̄)
−f(X, Ḡ)α(X̂)α( ˆḠ) + f(G, Ḡ)α(Ĝ)α( ˆḠ)]
= 𝔼(1 − v)(1 − v̄)[𝔼_{1}f(X, X̄)(X̂ ˆX̄)^{2}e{V + vX̂ + v̄ ˆX̄}

−2𝔼_{1}f(G, X̄)(Ĝ ˆX̄)^{2}e{V + vĜ + v̄ ˆX̄} + 𝔼_{1}f(G, Ḡ)(Ĝ ˆḠ)^{2}e{V + vĜ + v̄ˆḠ}].

From (4.42) and (4.57) we get for r = 1

|I_{1}| ≪ 𝔼|𝔼_{1}f(X, X̄)(X̂ ˆX̄)^{2}e{V + vX̂ + v̄ ˆX̄}|  (4.58)
+𝔼|𝔼_{1}f(G, X̄)(Ĝ ˆX̄)^{2}e{V + vĜ + v̄ ˆX̄}|
+𝔼|𝔼_{1}f(G, Ḡ)(Ĝ ˆḠ)^{2}e{V + vĜ + v̄ˆḠ}|.

We shall prove that in (4.58)

𝔼|f(X, X̄)|(X̂ ˆX̄)^{2} ≪ |t|^{5} \frac{1}{n^{2}} L_{n}.  (4.59)

This inequality is also valid for the expectations

𝔼|f(G, X̄)|(Ĝ ˆX̄)^{2} and 𝔼|f(G, Ḡ)|(Ĝ ˆḠ)^{2}.

Due to (4.46) and (a + b + c)^{2} ≤ 3(a^{2} + b^{2} + c^{2}) we write

𝔼|f(X, X̄)|(X̂ ˆX̄)^{2}  (4.60)
≪ \frac{|t|^{5}}{n^{5}} [𝔼|h(X, X̄)|𝔼_{1}(∑_{j=1}^{n−2}(h(X, X_{j})h(X̄, X_{j}) − 𝔼_{Z}h(X, Z)h(X̄, Z)))^{2}

+𝔼|h(X, X̄)|𝔼_{1}( ∑_{1≤i≠j≤n−2}(h(X, X_{i})h(X̄, X_{j}))^{2}

+n^{2}𝔼|h(X, X̄)|(𝔼_{Z}h(X, Z)h(X̄, Z))^{2}]

≪ \frac{|t|^{5}}{n^{5}} [n𝔼|h(X, X̄)|𝔼_{Z}h^{2}(X, Z)h^{2}(X̄, Z) + n^{2}𝔼|h(X, X̄)|𝔼_{Z}h^{2}(X, Z)𝔼Z̄h^{2}(X̄, Z̄)].

In (4.60), by (2.14) and (4.16),

𝔼|h(X, X̄)|𝔼_{Z}h^{2}(X, Z)h^{2}(X̄, Z) ≤ 4n𝔼_{Z}(𝔼_{X}h^{2}(X, Z))^{2}
≪ n𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{n}) + n(α_{n} + β_{n})^{2}

and by Hölder’s inequality

𝔼|h(X, X̄)|𝔼_{Z}h^{2}(X, Z)𝔼Z̄h^{2}(X̄, Z̄)  (4.61)
≪ 𝔼|h(X, X̄)|(∥X∥^{2}I(∥X∥ ≤ \sqrt{n}) + α_{n} + β_{n})(∥X̄∥^{2}I(∥X̄∥ ≤ \sqrt{n}) + α_{n} + β_{n})
≪ 𝔼∥X∥^{4}I(∥X∥ ≤ \sqrt{n}) + (α_{n} + β_{n})^{2}.

From (4.60) – (4.61) we get (4.59). Thus (4.40) for r = 1 follows from (4.56), (4.58), (4.59) and the partitioning argument for the sums X̂ in (4.58).

Verification of (4.40) for r = 2. In (4.42) by Taylor’s formula (4.8)

𝔼 exp{V }(φ_{2}(X, X̄) − 2φ_{2}(X, G) + φ_{2}(G, Ḡ))

= −𝔼(1 − v)[f^{2}(X, X̄)(X̂ + ˆ X̄)^{2}e{V + v(X̂ + ˆ X̄)}

−2f^{2}(X, G)(X̂ + Ĝ)^{2}e{V + v(X̂ + Ĝ)} + f^{2}(G, Ḡ)(Ĝ + ˆ Ḡ)^{2}e{V + v(Ĝ + ˆ Ḡ)}].

Hence in (4.42)

|I_{2}| ≪ 𝔼f^{2}(X, X̄)𝔼_{1}|(X̂ + ˆ X̄)^{2}e{V + v(X̂ + ˆ X̄)}|  (4.62)
+𝔼f^{2}(X, G)𝔼_{1}|(X̂ + Ĝ)^{2}e{V + v(X̂ + Ĝ)}|
+𝔼f^{2}(G, Ḡ)𝔼_{1}|(Ĝ + ˆ Ḡ)^{2}e{V + v(Ĝ + ˆ Ḡ)}|.

We will prove that

𝔼f^{2}(X, X̄) ˆX̄^{2} ≪ t^{4} \frac{1}{n^{2}} L_{n}.

Write, taking into account (4.16)

𝔼f^{2}(X, X̄) ˆX̄^{2} = 𝔼f^{2}(X, X̄)𝔼_{1} ˆX̄^{2}
= (n − 2)\frac{t^{4}}{n^{4}} 𝔼(𝔼X̄h^{2}(X, X̄)𝔼_{Z}h^{2}(X, Z))
≪ \frac{t^{4}}{n^{3}} 𝔼(∥X∥^{2} I(∥X∥ ≤ \sqrt{n}) + α_{n} + β_{n})^{2} ≪ t^{4} \frac{1}{n^{2}} L_{n}.

To get (4.40) from (4.62) it sufficies to partition the sums of type X̂ + ˆ X̄ into a sum of three parts with approximately equal numbers of summands, then conditioning to use the bound κ(t) and approximations of the type (4.59) and (4.61) for the expectations.

Verification of (4.40) for r = 3. In (4.42) by (4.8), the symmetry of G and the i.i.d. assumption

𝔼{V }(φ_{3}(X, X̄) − 2φ_{3}(X, G) + φ_{3}(G, Ḡ))
= 𝔼f^{3}(X, X̄)𝔼e{V } + 2i𝔼f^{3}(X, X̄)X̂ e{V + v(X̂ + ˆ X̄)}
−2i𝔼f^{3}(X, G)(X̂ + Ĝ) e{V + v(X̂ + Ĝ)}
+2i𝔼f^{3}(G, Ḡ)Ĝ e{V + v(Ĝ + ˆ Ḡ)}.

Therefore in (4.42)

|I_{3}| ≪ |𝔼f^{3}(X, X̄)|κ(t) + 𝔼|f(X, X̄)|^{3}|𝔼_{1}X̂e{V + v(X̂ + ˆ X̄)}| (4.63)
+|f(X, G)|^{3}|𝔼_{1}(X̂ + Ĝ)e{V + v(X̂ + Ĝ)}|
+𝔼|f(G, Ḡ)|^{3}|𝔼_{1}Ĝe{V + v(Ĝ + ˆ Ḡ)}|.

Here by (2.14)

|𝔼f^{3}(X, X̄)| ≪ |t|^{3} \frac{1}{n^{2}} (L_{n} + μ_{n}).

Partitioning the sum X̂ into a sum of two parts with approximately equal numbers of summands we find

𝔼|f(X, X̄)|^{3}|𝔼_{1}X̂e{V + v(X̂ + ˆ X̄)}|  (4.64)
≪ \sqrt{n}κ(t)𝔼|f(X, X̄)|^{3}(𝔼_{Z}f^{2}(X, Z))^{1/2}
= t^{4}κ(t)\frac{1}{n^{2}} n^{−3/2}𝔼|h(X, X̄)|^{3}(𝔼_{Z}h^{2}(X, Z))^{1/2}.

In (4.64) by the inequality ab ≤ a^{2} + b^{2}, (2.16) and (4.16)

n^{−3/2}𝔼|h(X, X̄)|^{3}(𝔼_{Z}h^{2}(X, Z))^{1/2}  (4.65)
= 𝔼\{(\frac{1}{n}𝔼X̄|h(X, X̄)|^{3})(\frac{1}{n}𝔼_{Z}h^{2}(X, Z))^{1/2}\}
≪ n^{−2}𝔼h^{4}(X, X̄) + n^{−1}𝔼h^{2}(X, X̄) ≪ L_{n}.

The other expectations in (4.63) are estimated similarly. From (4.63) – (4.65) we have (4.40) for r = 3. This completes the proof of (4.37).

Lemma 4.5. For t ∈ ℝ

|𝔼e{tU_{n}(g)} − 𝔼e{tU_{∞}(Φ)}| ≪ (|t| + t^{6})κ(t)(L_{n} + μ_{n}) + |t|n^{−3}L_{n}.

Proof. The result follows from (4.5) – (4.6), (4.9), (4.21), (4.31) – (4.33) and (4.37).

5 Proof of Theorem 1.1.

By Lemma 2.1.

Δ_{n} ≤ sup_{x} |ℙ{U_{n}(ϵ^{2}g) ≤ x} − ℙ{U_{∞}(ϵ^{2}Φ) ≤ x}| + α_{n} + β_{n}  (5.1)

with ϵ = (16s)^{−1}.

Choose s = 13. Due to Lemma 3.1 and the representations (4.1) and (4.3) we can apply Lemma 4.2 by Bentkus and Götze (1999) with m = m(13, q_{13}). This gives

sup_{x} |ℙ{U_{n}(ϵ^{2}g) ≤ x} − ℙ{U_{∞}(ϵ^{2}Φ) ≤ x}| ≪ J + R,  (5.2)

where

J = ∫_{−nt_{∗}}^{nt_{∗}} |𝔼e{tϵ^{2}U_{n}(g)} − 𝔼e{tϵ^{2}U_{∞}(Φ)}| |t|^{−1} dt

with t_{∗} = m^{−1}(pn/m)^{−1/2} and R ≪ (pn)^{−1}m. By our Lemma 4.5 and the estimate (4.20) in Bentkus and Götze (1999) we get

J ≪ p^{−6}(L_{n} + μ_{n}).  (5.3)

Collecting the bounds (5.1) – (5.3) we obtain (1.10). This proves Theorem 1.1.

REFERENCES

1. Bentkus, V. and Götze, F. (1999). Optimal bounds in non-Gaussian limit theorems for U-statistics. Ann. Probab. 27, 454-521.

2. Borovskikh, Yu.V. (1996). U-statistics in Banach Spaces, VSP, Utrecht, The Netherlands.

3. Dunford, N. and Schwartz, J. (1961). Linear Operators. Part II: Spectral Theory, Interscience Publishers, New York.

4. de la Peña, V.H. (1992). Decoupling and Khintchine’s inequalities for Ustatistics. Ann. Probab. 20, 1877-1892.