Sharp estimates of the rate of convergence for U-statistics

Yuri V.Borovskikh

August, 1998 Department of Applied Mathematics St.Petersburg Transport University

The rate of convergence for non-degenerate U−statistics of n i.i.d. random variables is investigated in the conditions of the central limit theorem. The basic result sharpens and improves earlier results of many papers devoted to this problem for such random sums. The Berry-Esséen bounds of order O(n^{−1/2}) hold for non-degenerate U−statistics of degree m with canonical Hoeffding representation

U_{n} − θ = \frac{m}{n} ∑_{i_{1}=1}^{n} g_{1}(X_{i_{1}}) + \frac{m(m − 1)}{n(n − 1)} ∑_{1≤i_{1}<i_{2}≤n} g_{2}(X_{i_{1}}, X_{i_{2}}) + etc.

under following minimal sufficient conditions on canonical functions g_{c}, c = 1, …, m : for c = 1 there exists absolute moment of order 3 of random variable g_{1}, i.e. E|g_{1}|^{3} < ∞ and for 2 ≤ c ≤ m there exist weak moments of order δ_{c} of random variables g_{2}, …, g_{m}, i.e. sup_{t≥0}(t^{δ_{c}}P (|g_{c}| > t)) < ∞ where δ_{c} = (2c + 1)/(2c − 1).

1. Introduction.

Let X_{1}, …, X_{n} be independent random variables taking values in a measurable space (𝔛, 𝔅), and all with an identical distribution P. Consider U−statistics (see Hoeffding (1948))

U_{n} = \binom{n}{m}^{−1} ∑_{1≤i_{1}<···<i_{m}≤n} Φ(X_{i_{1}}, …, X_{i_{m}})  (1)

with a symmetric kernel Φ : 𝔛^{m} ↦ R such that E|Φ(X_{1}, …, X_{m})| < ∞. Define for c = 0, 1, …, m the conditional expectations

Φ_{c}(x_{1}, …, x_{c}) = E Φ(x_{1}, …, x_{c}, X_{c+1}, …, X_{m})

AMS 1991 subject classification. Primary 60F05.

Key words and phrases. U−statistics, central limit theorem, rates of convergence.

setting Φ_{0} = E Φ(X_{1}, …, X_{m}) and Φ_{m} = Φ, and then define for c = 1, …, m the functions g_{c} : 𝔛^{c} ↦ R by the formula

g_{c}(x_{1}, …, x_{c}) = ∑_{d=1}^{c}(−1)^{c−d} ∑_{1≤j_{1}<···<j_{d}≤c}(Φ_{d}(x_{j_{1}}, …, x_{j_{d}}) − θ),

where θ = E Φ(X_{1}, …, X_{m}). The functions g_{c} possess the property of degeneracy E g_{c}(x_{1}, …, x_{c−1}, X_{c}) = 0, c = 1, …, m, and are called canonical. According to the canonical Hoeffding representation

U_{n} − θ = \frac{m}{n} ∑_{i_{1}=1}^{n} g_{1}(X_{i_{1}}) + \frac{m(m − 1)}{n(n − 1)} ∑_{1≤i_{1}<i_{2}≤n} g_{2}(X_{i_{1}}, X_{i_{2}})

+ ⋯ + \frac{m(m − 1) ⋯ 2 · 1}{n(n − 1) ⋯ (n − m + 1)} ∑_{1≤i_{1}<···<i_{m}≤n} g_{m}(X_{i_{1}}, …, X_{i_{m}}),  (2)

i.e. any U-statistic is a linear combination of pairwise uncorrelated U-statistical sums of the form (1) with degenerate kernels g_{c}, c = 1, …, m. Hoeffding (1948, 1961) developed a comprehensive theory of U-statistics. In particular, using the decomposition (2) he proved the asymptotic normality of normalized U-statistics. In the equation (2) the first sum m n^{−1}(g_{1}(X_{1}) + ⋯ + g_{1}(X_{n})) is known as the Hoeffding projection of the U-statistic, and is clearly the usual sum of independent identically distributed random variables.

In what follows we shall consider U-statistics with a symmetric non-degenerate kernel Φ : 𝔛^{m} → R. It is assumed that 0 < Eg_{1}^{2}(X_{1}) < ∞. Without loss of generality Eg_{1}^{2}(X_{1}) = 1 can be taken. Denote

F_{n}(x) := P (\sqrt{n} (U_{n} − θ)/m ≤ x), F (x) := P (τ ≤ x), x ∈ R,

where τ is a standard normal random variable. By the central limit theorem for nondegenerate U-statistics, as n → ∞ (see Arcones (1994), Borovskikh (1989, 1996))

Δ_{n} := sup_{x} |F_{n}(x) − F (x)| → 0,

if

t^{γ_{c}}P (|g_{c}| > t) → 0 as t → ∞

for each c = 2, …, m, where γ_{c} = 2c/(2c − 1). Note that

1 + \frac{1}{2m − 1} = γ_{m} < γ_{m−1} < ⋯ < γ_{3} < γ_{2} = \frac{4}{3} .

Here and below, it is assumed that g_{c} = g_{c}(X_{1}, …, X_{c}) for all c = 1, 2,…, m.

The problem of estimation of convergence rate in this central limit theorem was examined by many authors (see Borovskikh (1996), Lee (1990)). In every succeeding paper, conditions imposed on the kernel Φ were weakened or the order of decreasing the variables Δ_{n} to zero with n → ∞ was refined for different values of the parameter m. The first estimates of order O(n^{−l/(2(l+1))}) were obtained by Malevich and Abdalimov (1970) and Grams and Serfling (1973) under the assumption E|Φ|^{l} < ∞ as l ≥ 3, for any fixed m ≥ 2. Then, the case of m = 2 was under active investigation. For example, Bickel (1974) obtained an estimate of the convergence rate of the order O(n^{−1/2}) under the assumption that the kernel Φ is bounded. Chan and Wierman (1977) established that the estimate O(n^{−1/2}) holds true if Φ has a finite fourth moment; they also showed that if Φ only has finite third absolute moment, then the estimate O(n^{−1/2}(ln n)^{1/3}) holds. The estimate of the order O(n^{−1/2}) under the assumption E|Φ|^{3} < ∞ was obtained by Callaert and Janssen (1978); here, the authors follow the method of characteristic functions and develop the key idea of Chan and Wierman (1977) about a partition of the U-statistical sum, and moreover, employ martingale properties of U-statistics. Borovskikh (1979) weakened the condition E|Φ|^{3} < ∞ and proved the estimate O(n^{−1/2}) under the conditions E|g_{1}|^{3} < ∞ and EΦ^{2} < ∞, i.e., an absolute moment of the third order is required only for g_{1}. Helmers and van Zwet (1982) obtained the estimate O(n^{−1/2}) under the condition E|Φ|^{5/3+ε} < ∞ for ε > 0. Borovskikh and Korolyuk (1984, 1985) weakened this condition to E|Φ|^{5/3} < ∞. This result can also be found in Friedrich (1989), Bolthausen and Götze (1989), Lee (1990). Lower bounds of the order O(n^{−1/2}) for the convergence rate under conditions E|g_{1}|^{3} < ∞ and EΦ^{2} < ∞ (and for moments of higher orders) were obtained by Maesono (1991) (see also Maesono and Yamato (1994)). Bentkus, Götze and Zitikis (1994) constructed lower bounds in the case m = 2 under the conditions E|g_{1}|^{3} < ∞ and E|Φ|^{5/3−ε} < ∞ for ε > 0, and gave an example of kernel Φ and random variables X_{j} = X_{j}^{(n)}, j = 1, …, n, such that the estimate Δ_{n} ≥ c(ε) n^{−1/2+3ε/2} holds with a constant c(ε) > 0. Indeed, this estimate does not mean that the condition E|Φ|^{5/3} < ∞ is necessary and minimal for the estimate of order O(n^{−1/2}) to hold. Bentkus and Götze (1995) generalized this result to the case of arbitrary m ≥ 2 with analogous lower bounds for Δ_{n}.

2. Results.

We shall consider the problem of estimating of the convergence rate both under general conditions Eg_{1}^{2}(X_{1}) = 1 and E|Φ(X_{1}, …, X_{m})| < ∞, and under minimal conditions imposed on canonical functions g_{c}, c = 1,…, m, which are sufficient for the estimate O(n^{−1/2}) with any natural m ≥ 1. In what follows, we shall need the following key functions.

1. Truncated canonical functions

g_{c}^{n}: (x_{1}, …, x_{c}) → g_{c}^{n}(x_{1}, …, x_{c}), c = 1, …, m,

where

g_{c}^{n}(x_{1}, …, x_{c}) = g_{c}(x_{1}, …, x_{c}) I(|g_{c}(x_{1}, …, x_{c})| ≤ n^{(2c−1)/2}).

Here n ≥ 2 and I(A) is the indicator function of a set A ∈ 𝔅.

2. Truncated moment functions

β_{n} = ∑_{c=2}^{m} \binom{m}{c}\{\sqrt{n} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\},
γ_{n}(p) = ∑_{c=2}^{m} \binom{m}{c}\{n^{(2c−p(2c−1))/2} E|g_{c}|^{p} I(|g_{c}| ≤ n^{(2c−1)/2})\},

where 1 ≤ p ≤ 2.

3. Weak moment functions

B_{c}(x, p) = sup_{t≥x}(t^{p}P (|g_{c}| > t)), c = 1, …, m,

where x ≥ 0, and p > 0.

4. Weak moments of order p

B_{c}(0, p) = sup_{t≥0}(t^{p}P (|g_{c}| > t)), c = 1, …, m.

Theorem 1. Let the following conditions hold

Eg_{1}^{2}(X_{1}) = 1, E|Φ(X_{1}, …, X_{m})| < ∞.

Then for all m ≥ 1, n ≥ m and any p, 1 ≤ p < 2,

Δ_{n} ≤ 132.8 E|g_{1}|^{3} I(|g_{1}| ≤ \sqrt{n}) · \frac{1}{\sqrt{n}} + 2.85 Eg_{1}^{2} I(|g_{1}| > \sqrt{n})

+\sqrt{6π} e^{m/3} 6^{m/2}m! ∑_{c=2}^{m} \binom{m}{c} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}

+2^{2m^{2}+4m}6^{m}m^{2m}m! 3^{6/(2−p)}Γ(6/(2 − p)) · {β_{n} + γ_{n}(p)},  (3)

where Γ(z) is the gamma function.

Estimate (3) holds true under the conditions of the central limit theorem, too, i.e., (3) implies the following statement.

Corollary 1. If Eg_{1}^{2}(X_{1}) = 1, and B-conditions

B_{c}(x, γ_{c}) → 0, c = 2, …, m,

hold as x → ∞, then

sup_{x} |F_{n}(x) − F (x)| → 0 as n → ∞.

It appears that the weak moment B-functions not only express sufficient conditions for the central limit theorem for U-statistics to hold, but moreover, they control the rate of this convergence. Actually, the following theorem holds involving weak moments B_{c}(0, δ_{c}) of order δ_{c}, where

δ_{c} = (2c + 1)/(2c − 1), c = 1, …, m.

Note that

1 + \frac{2}{2m − 1} = δ_{m} < δ_{m−1} < ⋯ < δ_{3} < δ_{2} = \frac{5}{3} .

Theorem 2. Let the moment conditions

Eg_{1}^{2}(X_{1}) = 1, E|g_{1}(X_{1})|^{3} < ∞,

and B-conditions

B_{c}(0, δ_{c}) < ∞, c = 2, …, m,

be satisfied. Then for all m ≥ 1 and n ≥ m, the inequality

Δ_{n} ≤ A_{m}\{∑_{c=1}^{m} \binom{m}{c}(E|g_{c}| ∏_{s=1}^{c} |g_{1}(X_{s})|^{2/c} + B_{c}(0, δ_{c}))\} · \frac{1}{\sqrt{n}}  (4)

holds with some positive constant A_{m} which depends only on m, in addition,

E|g_{c}| ∏_{s=1}^{c} |g_{1}(X_{s})|^{2/c} ≤ (E|g_{1}|^{3})^{c} + 3cB_{c}(0, δ_{c}).

Particularly, in the case of m = 2, for all n ≥ 2

Δ_{n} ≤ \{215.6 E|g_{1}|^{3} + 46.2 (E|g_{1}|^{3}E|g_{2}|^{3/2})^{2/3} + 412940 B_{2}(0, 5/3)\} · \frac{1}{\sqrt{n}},

where

E|g_{2}|^{3/2} ≤ min{1 + 9B_{2}(0, 5/3), 10 B_{2}^{9/10}(0, 5/3)}.

Corollary 2. Let Eg_{1}^{2}(X_{1}) = 1 and the moment conditions

E |g_{c}|^{δ_{c}} < ∞, c = 1, …, m

be satisfied. Then for all m ≥ 1 and n ≥ m

Δ_{n} ≤ A_{m}\{ ∑_{c=1}^{m} \binom{m}{c} E |g_{c}|^{δ_{c}} \} · \frac{1}{\sqrt{n}}

with some positive constant A_{m} which depends only on m. In particularly, if m = 2 then for all n ≥ 2

Δ_{n} ≤ \{233.88 E|g_{1}|^{3} + 412967.72 E|g_{2}|^{5/3}\} · \frac{1}{\sqrt{n}}.

Theorem 3. Let the moment conditions

Eg_{1}^{2}(X_{1}) = 1, E|g_{1}(X_{1})|^{3} < ∞,

and B-conditions

B_{c}(0, δ_{c}) < ∞, c = 2, …, m,

hold. Then for all m ≥ 1 and n ≥ m

Δ_{n} ≤ A_{m}\{∑_{c=1}^{m} \binom{m}{c}(E|g_{c}| ∏_{s=1}^{c} |g_{1}(X_{s})|^{2/c} + B_{c}(n^{1/8}, δ_{c}))\} · \frac{1}{\sqrt{n}}  (5)

with some positive constant A_{m} which depends only on m.

Note that B_{c}(n^{1/8}, δ_{c}) ≤ B_{c}(0, δ_{c}), c = 1,…, m.

Theorem 4. Let the moment conditions

Eg_{1}^{2}(X_{1}) = 1, E|g_{1}(X_{1})|^{3} < ∞,

and B-conditions

B_{c}(0, δ_{c}) < ∞, c = 2, …, m,

hold. Then for all m ≥ 1 and n ≥ m the following inequality holds

Δ_{n} ≤ A_{m} \{E|g_{1}|^{3} + 3(m − 1) E|g_{2}(X_{1}, X_{2}) g_{1}(X_{1}) g_{1}(X_{2})| + ρ_{n}\} · \frac{1}{\sqrt{n}}, (6)

where

ρ_{n} = ∑_{c=2}^{m} B_{c}(n^{1/8}, δ_{c}) + ∑_{c=3}^{m} B_{c}^{2/3}(0, δ_{c})(E|g_{1}|^{3} + ∑_{d=2}^{m} B_{d}(n^{1/8}, δ_{d}))^{c/3} · n^{−(c−1)/6},

and A_{m} > 0 is a constant which depends only on m.

Here ρ_{n} → 0 as n → ∞, provided that B_{c}(x, δ_{c}) → 0, c = 2,…, m, as x → ∞. On asymptotic expansion. If the Cramer condition for the random variable

g_{1}(X_{1}) is assumed in addition to the conditions of Theorem 4, then the estimate (6) can be improved. Denote

κ_{3} = Eg_{1}^{3}(X_{1}) + 3(m − 1) E(g_{2}(X_{1}, X_{2}) g_{1}(X_{1}) g_{1}(X_{2})),
χ(t) = E exp(it g_{1}(X_{1})), t ∈ R.

Theorem 5. Let the moment conditions

Eg_{1}^{2}(X_{1}) = 1, E|g_{1}(X_{1})|^{3} < ∞,

the Cramer condition

lim_{|t|→∞} sup |χ(t)| < 1,

and B-conditions

B_{c}(x, δ_{c}) → 0, c = 2, …, m,

as x → ∞ hold. Then the following relation holds as n → ∞

sup_{x}|F_{n}(x) − F (x) − \frac{κ_{3}}{6} · \frac{1}{\sqrt{2πn}} (1 − x^{2}) e^{−x^{2}/2}| = o(n^{−1/2}).  (7)

Theorem 5 can be found in Maesono and Yamato (1994) with condition E|Φ|^{3} < ∞ instead of B-conditions B_{c}(x, δ_{c}) → 0, c = 2,…, m as x → ∞.

Comparison of conditions. Concerning B-conditions for which (4) holds, note the following inequalities which are always true:

B_{c}(0, δ_{c}) ≤ E|g_{c}|^{δ_{c}}, c = 1, …, m.

The inverse inequalities are not true in general. We can only assert that for any positive p, p < δ_{c},

E|g_{c}|^{p} ≤ δ_{c}(δ_{c} − p)^{−1}B_{c}^{p/δ_{c}}(0, δ_{c}).

Example 1. Assume that the random variable g_{c} has the probability density

p(x) = δ_{c}(2(1 + δ_{c}))^{−1} min{1, |x|^{−δ_{c}−1}}, x ∈ R.

and, hence, x^{δ_{c}}P (|g_{c}| > x) = (1 + δ_{c})^{−1}, x ≥ 1.

Then for all c = 2, …, m

B_{c}(0, δ_{c}) = (1 + δ_{c})^{−1} and E|g_{c}|^{δ_{c}} = ∞.

Example 2. Suppose that g_{c} is δ_{c}−stable random variable such that for any t ∈ R

E exp(itg_{c}) = exp(−|t|^{δ_{c}}), c = 2, … m

It is well-known that in this situation for all x ≥ 2

α_{1} ≤ x^{δ_{c}}P (|g_{c}| > x) ≤ α_{2}

and

α_{3} ln x ≤ E|g_{c}|^{δ_{c}}I(1 ≤ |g_{c}| ≤ x) ≤ α_{4} ln x

with some positive constants α_{j}.

That is

α_{1} ≤ B_{c}(0, δ_{c}) ≤ α_{2}, but E|g_{c}|^{δ_{c}} = ∞

for all c = 2, …, m.

Therefore, the B-conditions B_{c}(0, δ_{c}) < ∞, c = 2, …, m, of Theorem 2 are weaker then the moment conditions E|g_{c}|^{δ_{c}} < ∞, c = 2,…, m, of Corollary 2.

3. Proofs.

Theorems 2, 3 and 4 are essentially consequences of Theorem 1, since we derive estimates (4)–(6) from (3). The proof of Theorem 5 uses also the estimates of characteristic functions, obtained in the proof of Theorem 1. In turn, the proof of Theorem 1 is based on Hoeffding decomposition, on truncated inequalities and method of characteristic functions. Analysing the characteristic function of a normalized U-statistic with truncated canonical functions g_{c}^{n}, c = 1,…, m, we develop the idea of Chan and Wierman (1977) on a partition of U-statistical sum and apply martingale structure of U-statistics (see Borovskikh and Korolyuk (1997)).

Proof of Theorem 1. The following Lemma contains general truncated inequality, which enables one to introduce truncated random variables g_{c}^{n}(X_{1}, …, X_{c}) instead of g_{c}(X_{1}, …, X_{c}), c = 1, …, m. For this purpose, first denote

S_{nm}(Φ) = ∑_{1≤i_{1}<···<i_{m}≤n} Φ(X_{i_{1}}, …, X_{i_{m}}) + f(X_{1}, …, X_{n}),

where Φ: 𝔛^{m} → R is a symmetric function of m variables, f: 𝔛^{n} → R is an arbitrary measurable function of n variables. Let Φ^{t} denote the truncated at the level t > 0 function Φ i.e.,

Φ^{t}(x_{1}, …, x_{m}) = Φ(x_{1}, …, x_{m}) I(|Φ(x_{1}, …, x_{m})| ≤ t)

with the indicator function I(A) of a set A ∈ 𝔅.

Lemma 1. For all x ∈ R and any t > 0

|P (S_{nm}(Φ) ≤ x) − P (S_{nm}(Φ^{t}) ≤ x)|
≤ \frac{n(n − 1) ⋯ (n − m + 1)}{m!} P (|Φ(X_{1}, …, X_{m})| > t).  (8)

Proof. First, consider the case m = 2. For each k = 1, …, n, denote

α_{k}(Φ) = ∑_{i=k+1}^{n} Φ(X_{k}, X_{i}),

ζ_{k} = ∑_{j=1}^{k−1} ∑_{i=j+1}^{n} Φ^{t}(X_{i}, X_{j}) + ∑_{j=k+1}^{n−1} ∑_{i=j+1}^{n} Φ(X_{i}, X_{j}) + f(X_{1}, …, X_{n}),

and put

θ_{l} = ∑_{i<l} Φ(X_{k}, X_{i}) + ∑_{i>l} Φ^{t}(X_{k}, X_{i})

for l = k + 1,…, n. The following formulas hold

P (S_{n2}(Φ) ≤ x) − P (S_{n2}(Φ^{t}) ≤ x)
= E{I(S_{n2}(Φ) ≤ x) − I(S_{n2}(Φ^{t}) ≤ x)}
= E{I(α_{1}(Φ) + ζ_{1} ≤ x) − I(α_{1}(Φ^{t}) + ζ_{1} ≤ x)}
+ E{I(α_{2}(Φ) + ζ_{2} ≤ x) − I(α_{2}(Φ^{t}) + ζ_{2} ≤ x)}
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ E{I(α_{n−1}(Φ) + ζ_{n−1} ≤ x) − I(α_{n−1}(Φ^{t}) + ζ_{n−1} ≤ x)}
= ∑_{k=1}^{n−1} E{I(α_{k}(Φ) + ζ_{k} ≤ x) − I(α_{k}(Φ^{t}) + ζ_{k} ≤ x)}.

By similar arguments, we get the formula

E(I(α_{k}(Φ) + ζ_{k} ≤ x) − I(α_{k}(Φ^{t}) + ζ_{k} ≤ x))

= ∑_{l=k+1}^{n} E(I(Φ(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x) − I(Φ^{t}(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x)).

Therefore,

P (S_{2n}(Φ) ≤ x) − P (S_{2n}(Φ^{t}) ≤ x)

= ∑_{k=1}^{n−1} ∑_{l=k+1}^{n} E{I(Φ(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x) − I(Φ^{t}(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x)}.

Here

EI(Φ^{t}(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x)
= EI(Φ^{t}(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x) I(|Φ(X_{k}, X_{l})| ≤ t)
+EI(Φ^{t}(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x) I(|Φ(X_{k}, X_{l})| > t)
= EI(Φ(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x)
−EI(Φ(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x) I(|Φ(X_{k}, X_{l})| > t)
+EI(θ_{l} + ζ_{k} ≤ x) I(|Φ(X_{k}, X_{l})| > t).

Since

|I(Φ(X_{k}, X_{l}) + θ_{k} + ζ_{k} ≤ x) − I(θ_{l} + ζ_{k} ≤ x)| ≤ 1,

we have as a result the following inequality

|E(I(Φ(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x) − I(Φ^{t}(X_{k}, X_{l}) + θ_{l} + ζ_{k} ≤ x))|

≤ P (|Φ(X_{1}, X_{2})| > t),

which is valid for any t > 0.

Consequently, for all x ∈ R and t > 0

|P (S_{n2}(Φ) ≤ x) − P (S_{n2}(Φ^{t}) ≤ x)| ≤ \frac{n(n − 1)}{2} P (|Φ(X_{1}, X_{2})| > t),

which proves (8) for m = 2.

Now let us consider the general case m ≥ 2. According to the definition

S_{nm}(Φ) = ∑_{i_{1}=1}^{n−m+1} ∑_{i_{2}=i_{1}+1}^{n−m+2} … ∑_{i_{m}=i_{m−1}+1}^{n} Φ(X_{i_{1}}, X_{i_{2}}, …, X_{i_{m}}) + f(X_{1}, …, X_{n}).

For all integer numbers r_{1}, r_{2},…, r_{c}, such that 1 ≤ r_{1} < r_{2} < … < r_{c} ≤ n−m+c, c = 1, …, m, denote

α_{r_{1}r_{2}...r_{c}}(Φ) = ∑_{i_{c+1}=r_{c}+1}^{n−m+c+1} … ∑_{i_{m}=i_{m−1}+1}^{n} Φ(X_{r_{1}}, X_{r_{2}}, …, X_{r_{c}}, X_{i_{c+1}} …, X_{i_{m}}),

so that α_{r_{1}r_{2}...r_{m}}(Φ) = Φ(X_{r_{1}}, X_{r_{2}}, …, X_{r_{m}});

ζ(r_{1}, r_{2}, …, r_{c})

= ∑_{i_{c}=r_{c−1}+1}^{r_{c}−1} ∑_{i_{c+1}=i_{c}+1}^{n−m+c+1} … ∑_{i_{m}=i_{m−1}+1}^{n} Φ^{t}(X_{r_{1}}, X_{r_{2}}, …, X_{r_{c−1}}, X_{i_{c}}, …, X_{i_{m}})

+ ∑_{i_{c}=r_{c}+1}^{n−m+c} ∑_{i_{c+1}=i_{c}+1}^{n−m+c+1} … ∑_{i_{m}=i_{m−1}+1}^{n} Φ(X_{r_{1}}, X_{r_{2}}, …, X_{r_{c−1}}, X_{i_{c}}, …, X_{i_{m}}),

and

ζ_{c} = ∑_{s=1}^{c} ζ(r_{1}, r_{2}, …, r_{s}) + f(X_{1}, …, X_{n}).

In these notations, for all c = 1, …, m, the following formulas are valid

P (S_{nm}(Φ) ≤ x) − P (S_{nm}(Φ^{t}) ≤ x)

= ∑_{r_{1}=1}^{n−m+1} ∑_{r_{2}=r_{1}+1}^{n−m+2} … ∑_{r_{c}=r_{c−1}+1}^{n−m+c} E{I(α_{r_{1}...r_{c}}(Φ) + ζ_{c} ≤ x) − I(α_{r_{1}...r_{c}}(Φ^{t}) + ζ_{c} ≤ x)}

= ∑_{1≤r_{1}<r_{2}<...<r_{c}≤n−m+c} E{I(α_{r_{1}...r_{c}}(Φ) + ζ_{c} ≤ x) − I(α_{r_{1}...r_{c}}(Φ^{t}) + ζ_{c} ≤ x)}.

Put c = m here. Then

P (S_{nm}(Φ) ≤ x) − P (S_{nm}(Φ^{t}) ≤ x)

= ∑_{1≤r_{1}<...<r_{m}≤n} E{I(α_{r_{1}...r_{m}}(Φ) + ζ_{m} ≤ x) − I(α_{r_{1}...r_{m}}(Φ^{t}) + ζ_{m} ≤ x)}.

Since

|E{I(α_{r_{1}...r_{m}}(Φ) + ζ_{m} ≤ x) − I(α_{r_{1}...r_{m}}(Φ^{t}) + ζ_{m} ≤ x)}| ≤ P (|Φ(X_{1}, …, X_{m})| > t),

we get the proof of Lemma 1 for any m ≥ 2.

In what follows, we shall apply the truncated inequality (8). Besides, we shall need the following representation of the kernel Φ(x_{1}, … x_{m}) via canonical functions g_{c}:

Φ(x_{1}, …, x_{m}) = θ + ∑_{c=1}^{m} ∑_{1≤i_{1}<···<i_{c}≤m} g_{c}(x_{i_{1}}, …, x_{i_{c}}),

just as the inverse representation of canonical functions g_{c} via the kernel Φ

g_{c}(x_{1}, …, x_{c}) = ∫_{X} … ∫_{X} Φ(y_{1}, … y_{m}) ∏_{s=1}^{c}(δ_{x_{s}}(dy_{s}) − P (dy_{s})) ∏_{s=c+1}^{m} P (dy_{s}), (9)

and the Hoeffding representation (2) in the form

U_{n}(Φ) − θ = ∑_{c=1}^{m} \binom{m}{c} U_{n}(g_{c}).

Then, we introduce the notations

g_{nc}(x_{1}, …, x_{c}) = ∫_{X} … ∫_{X} g_{c}(y_{1}, … y_{c}) I(|g_{c}| ≤ n^{(2c−1)/2}) ∏_{s=1}^{c}(δ_{x_{s}}(dy_{s}) − P (dy_{s})),

c = 1, 2, …, m;

g_{ncd}(x_{1}, …, x_{d}) = − ∫_{X} ∫_{X} g_{c}(y_{1}, …, y_{c}) I(|g_{c}| > n^{(2c−1)/2})

× ∏_{s=1}^{d}(δ_{x_{s}}(dy_{s}) − P (dy_{s})) · ∏_{s=d+1}^{c} P (dy_{s}), c ≥ d + 1;

h_{c} = g_{nc1},  h_{c}^{n} = h_{c} I(|h_{c}| ≤ \sqrt{n}),
h_{nc} = h_{c}^{n} − Eh_{c}^{n}, c = 2, 3, …, m;
S_{n} = ξ_{1} + ⋯ + ξ_{n},

ξ_{j} = \frac{1}{σ_{n}} · \frac{1}{\sqrt{n}} (g_{n1}(X_{j}) + ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} h_{nc}(X_{j})), j = 1, …, n;

σ_{n}^{2} = E(g_{n1}(X_{1}) + ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} h_{nc}(X_{1}))^{2};

v_{n} = \frac{\sqrt{n}}{σ_{n}m} · ∑_{c=2}^{m} \binom{m}{c} U_{n}(g_{nc}),

w_{n} = \frac{\sqrt{n}}{σ_{n}m} ∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d} U_{n}(g_{ncd}),

so that w_{n} = 0 as m = 1, 2.

Lemma 2. The inequality holds

sup_{x}|P (\sqrt{n} (U_{n} − θ)/m ≤ x) − F (x)|
≤ sup_{x}|P (S_{n} + v_{n} + w_{n} ≤ x) − F (x)|

+(1 + \frac{1}{\sqrt{2π}} ) E|g_{1}|^{2} I(|g_{1}| > \sqrt{n})

+3 · β_{n} + \frac{1}{\sqrt{2πe}} |σ_{n}^{2} − 1|/σ_{n}.  (10)

Proof. First apply Lemma 1 (inequality (8)) m times: at the first step we introduce random variables g_{1}^{n}(X_{i_{1}}); at the second step introduce random variables g_{2}^{n}(X_{i_{1}}, X_{i_{2}}), i_{1} < i_{2}, etc., at the m-th step introduce random variables g_{m}^{n}(X_{i_{1}}, …, X_{i_{m}}), i_{1} < ⋯ < i_{m}; then we obtain the inequality

sup_{x}|P (\sqrt{n} (U_{n} − θ)/m ≤ x) − P (\frac{\sqrt{n}}{m} · ∑_{c=1}^{m} \binom{m}{c} U_{n}(g_{c}^{n}) ≤ x)|

≤ ∑_{c=1}^{m} \frac{1}{c!} n^{[c]} P (|g_{c}| > n^{(2c−1)/2}).  (11)

Write the Hoeffding decomposition for the U-statistic with kernel the g_{c}^{n}:

U_{n}(g_{c}^{n}) = θ_{c}^{n} + ∑_{d=1}^{c} \binom{c}{d} U_{n}(g_{ncd}),

where

θ_{c}^{n} = Eg_{c}(X_{1}, …, X_{c}) I(|g_{c}| ≤ n^{(2c−1)/2}),
g_{ncd}(x_{1}, …, x_{d}) = ∫_{X} … ∫_{X} g_{c}(y_{1}, …, y_{c}) I(|g_{c}| ≤ n^{(2c−1)/2})

× ∏_{s=1}^{d}(δ_{x_{s}}(dy_{s}) − P (dy_{s})) ∏_{s=d+1}^{c} P (dy_{s}).

Here g_{ncd} = g_{ncc} = g_{nc} for c = d. Denote

θ̄_{c}^{n} := −Eg_{c}(X_{1}, …, X_{c}) I(|g_{c}| > n^{(2c−1)/2}).

Also by the degeneracy property of g_{c}(x_{1}, …, x_{c}), we see that θ_{c}^{n} = θ̄_{c}^{n}. Therefore

∑_{c=1}^{m} \binom{m}{c} U_{n}(g_{c}^{n}) = ∑_{c=1}^{m} \binom{m}{c} θ̄_{c}^{n} + ∑_{c=1}^{m} \binom{m}{c} ∑_{d=1}^{c} \binom{c}{d} U_{n}(g_{ncd})

= ∑_{c=1}^{m} \binom{m}{c} θ̄_{c}^{n} + ∑_{c=1}^{m} \binom{m}{c}\binom{c}{1} U_{n}(g_{nc1})

+ ∑_{c=2}^{m} \binom{m}{c} U_{n}(g_{nc}) + ∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d} U_{n}(g_{ncd}).

In this formula

∑_{c=1}^{m} \binom{m}{c}\binom{c}{1} U_{n}(g_{nc1}) = \frac{m}{n} · ∑_{j=1}^{n} g_{n1}(X_{j}) + ∑_{c=2}^{m} \binom{m}{c} c · \frac{1}{n} ∑_{j=1}^{n} h_{c}(X_{j}).

Therefore, in (11)

\frac{\sqrt{n}}{m} · ∑_{c=1}^{m} \binom{m}{c} U_{n}(g_{c}) = \frac{\sqrt{n}}{m} ∑_{c=1}^{m} \binom{m}{c} θ̄_{c}^{n} + \frac{1}{\sqrt{n}} ∑_{j=1}^{n} g_{n1}(X_{j})

+ ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} · \frac{1}{\sqrt{n}} ∑_{j=1}^{n} h_{c}(X_{j}) + (v_{n} + w_{n}) · σ_{n}.

Using this formula and Lemma 1, we introduce the truncated random variables h_{c}^{n}(X_{j}), j = 1,…, n, i.e., we get the inequality

sup_{x}|P (\frac{\sqrt{n}}{m} · ∑_{c=1}^{m} \binom{m}{c} U_{n}(g_{c}^{n}) ≤ x) − P (∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} · \frac{1}{\sqrt{n}} ∑_{j=1}^{n} h_{c}^{n}(X_{j}) + f_{n} ≤ x)|

≤ n ∑_{c=2}^{m} P (|h_{c}(X_{1})| > \sqrt{n}),  (12)

where

f_{n} = \frac{\sqrt{n}}{m} ∑_{c=1}^{m} \binom{m}{c} θ̄_{c}^{n} + \frac{1}{\sqrt{n}} ∑_{j=1}^{n} g_{n1}(X_{j})

+ ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} · \frac{1}{\sqrt{n}} ∑_{j=1}^{n} h_{c}^{n}(X_{j}) + (v_{n} + w_{n}) · σ_{n}.

In (12) we have the following representation

∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} \frac{1}{\sqrt{n}} ∑_{j=1}^{n} h_{c}^{n}(X_{j}) + f_{n} = σ_{n} (S_{n} + v_{n} + w_{n}) + η_{n},

where

η_{n} = \frac{\sqrt{n}}{m} ∑_{c=1}^{m} \binom{m}{c} θ̄_{c}^{n} + \sqrt{n} ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} Eh_{c}^{n}(X_{1}).

Join (11) and (12) taking into account the latter representation; we get

sup_{x}|P (\sqrt{n} (U_{n} − θ)/m ≤ x) − F (x)|

≤ sup_{x}|P (σ_{n}(S_{n} + v_{n} + w_{n}) + η_{n} ≤ x) − F (x)| + d_{n}
= sup_{x}|P (S_{n} + v_{n} + w_{n} ≤ x) − F (xσ_{n} + η_{n})| + d_{n},

where

d_{n} = n · ∑_{c=2}^{m} P (|h_{c}(X_{1})| > \sqrt{n}) + ∑_{c=1}^{m} \frac{1}{c!} n^{[c]} P (|g_{c}| > n^{(2c−1)/2}).

This yields

sup_{x}|P (\sqrt{n} (U_{n} − θ)/m ≤ x) − F (x)| ≤ sup_{x}|P (S_{n} + v_{n} + w_{n} ≤ x) − F (x)|

+ sup_{x} |F (x + η_{n}) − F (x)| + sup_{x} |F (σ_{n}x) − F (x)| + d_{n},

where, by the properties of the function F (x),

sup_{x} |F (x + η_{n}) − F (x)| ≤ \frac{1}{\sqrt{2π}} |η_{n}|,

and

sup_{x} |F (σ_{n}x) − F (x)| ≤ \frac{1}{\sqrt{2πe}} |σ_{n}^{2} − 1|/σ_{n}.

It remains to note that

d_{n} ≤ ∑_{c=1}^{m} \frac{1}{c!} \{\sqrt{n} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\}

+ ∑_{c=2}^{m}\{\sqrt{n} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\} ≤ 2β_{n},

|η_{n}| ≤ \frac{1}{m} ∑_{c=1}^{m} \binom{m}{c}\{\sqrt{n} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\}

+ ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} \{\sqrt{n} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\} ≤ 2β_{n}.

Lemma 2 is proved, since

1 + \frac{1}{c!} + \frac{1}{\sqrt{2π}} · \frac{1}{m}\binom{m}{c} + \frac{1}{\sqrt{2π}} \binom{m}{c}\frac{c}{m} ≤ 3\binom{m}{c}

for 2 ≤ c ≤ m, and moreover,

\sqrt{n} E|g_{1}| I(|g_{1}| > \sqrt{n}) ≤ Eg_{1}^{2}I(|g_{1}| > \sqrt{n}).

Our further aim is to estimate the first term on the right-hand side of (10) using the Esseen inequality. Denote by φ(t) the characteristic function of the random variable g_{n1}(X_{1}) + h̄_{n}(X_{1}), where h̄_{n}(X_{1}) = ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} h_{nc}(X_{1}), and denote by φ_{n}(t) the characteristic function of the random variable S_{n} + v_{n} + w_{n}. Therefore, φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) is the characteristic function of S_{n}. Then, by the Esseen inequality we have in (10)

sup_{x} |P (S_{n} + v_{n} + w_{n} ≤ x) − F (x)|

≤ \frac{1}{π} ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} |φ_{n}(t) − e^{−t^{2}/2}| |t|^{−1} dt + \frac{24 · a}{π\sqrt{2π}} · n^{−1/2}

≤ \frac{1}{π} ∫_{−\sqrt{n}/a}^{\sqrt{n}/a}|φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − e^{−t^{2}/2}| |t|^{−1} dt + \frac{24 · a}{π\sqrt{2π}} · n^{−1/2}

+ \frac{1}{π} ∫_{−\sqrt{n}/a}^{\sqrt{n}/a}|φ_{n}(t) − φ^{n}(\frac{t}{σ_{n}\sqrt{n}})| |t|^{−1} dt,  (13)

where

a = σ_{n}^{−3} E|g_{n1}(X_{1}) + h̄_{n}(X_{1})|^{3}.

First, we estimate the integral on the right-hand side of (13), which contains the difference φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − e^{−t^{2}/2}. Expanding the exponent into the Taylor series, we get

|E exp\{\frac{it}{σ_{n}\sqrt{n}} (g_{n1}(X_{1}) + h̄_{n}(X_{1}))\}| ≤ 1 − \frac{t^{2}}{2n} + \frac{a · |t|^{3}}{6 · n^{3/2}} .

Therefore, in the domain |t| ≤ \sqrt{n}/a

|φ(\frac{t}{σ_{n}\sqrt{n}})| ≤ exp(−\frac{t^{2}}{3n}).  (14)

Further, putting ψ(\frac{t}{\sqrt{n}}) = exp(−\frac{t^{2}}{2n}), we have

|φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − e^{−t^{2}/2}| = |φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − ψ^{n}(\frac{t}{\sqrt{n}})| = |φ(\frac{t}{σ_{n}\sqrt{n}}) − ψ(\frac{t}{\sqrt{n}})|

× |φ^{n−1}(\frac{t}{σ_{n}\sqrt{n}}) + φ^{n−2}(\frac{t}{σ_{n}\sqrt{n}})ψ(\frac{t}{\sqrt{n}}) + … + ψ^{n−1}(\frac{t}{\sqrt{n}})|.

Here, for all t ∈ R,

|φ(\frac{t}{σ_{n}\sqrt{n}}) − ψ(\frac{t}{\sqrt{n}})| ≤ |φ(\frac{t}{σ_{n}\sqrt{n}}) − 1 + \frac{t^{2}}{2n}| + |ψ(\frac{t}{\sqrt{n}}) − 1 + \frac{t^{2}}{2n}|

≤ \frac{1}{6} a · |t|^{3} · n^{−3/2} + \frac{1}{8} t^{4} n^{−2}.

Therefore, taking into account (14) with |t| ≤ \sqrt{n}/a, we have

|φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − e^{t^{2}/2}| ≤ (\frac{a · |t|^{3}}{6 · n^{3/2}} + \frac{t^{4}}{8n^{2}} ) ∑_{k=0}^{n−1} exp\{−\frac{(n − 1 − k)}{3n} t^{2} − \frac{k}{2n} t^{2}\}.

In addition, in the domain |t| ≤ \sqrt{n}/a we can write

∑_{k=0}^{n−1} exp\{−\frac{(n − 1 − k)}{3n} t^{2} − \frac{k}{2n} t^{2}\} = e^{− \frac{(n−1)}{3n} t^{2}} ∑_{k=0}^{n−1} exp\{−\frac{t^{2}}{6n} k\}

= e^{− \frac{(n−1)}{3n} t^{2}} \frac{(1 − exp(−\frac{t^{2}}{6} ))}{(1 − exp(−\frac{t^{2}}{6n}))} = exp\{\frac{t^{2}}{2n} − \frac{t^{2}}{3} \}\frac{(1 − exp(−\frac{t^{2}}{6} ))}{(exp(\frac{t^{2}}{6n}) − 1)}

≤ 6 e^{1/2} ne^{−t^{2}/3} \frac{1 − exp(−\frac{t^{2}}{6} )}{t^{2}}.

Thus, in the domain |t| ≤ \sqrt{n}/a the following inequality holds

|φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − e^{−t^{2}/2}| ≤ e^{1/2} · \frac{7}{4} · \frac{a}{\sqrt{n}} · |t| · e^{−t^{2}/3}.  (15)

Applying (15), we get the estimate for (13)

\frac{1}{π} ∫_{−\sqrt{n}/a}^{\sqrt{n}/a}|φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − e^{−t^{2}/2}| |t|^{−1} dt ≤ \frac{7}{4} · \sqrt{\frac{3e}{π}} · \frac{a}{\sqrt{n}};  (16)

here, according to the definition of a,

a ≤ 8 · σ_{n}^{−3} E|g_{1}^{n}(X_{1})|^{3} + 2^{2m+6} · σ_{n}^{−3} · β_{n} · \sqrt{n}.  (17)

Now we proceed with the estimating of the second integral on the right-hand side of (13). First, consider the characteristic function

φ_{n}(t) = E exp{itS_{n} + itv_{n} + itw_{n}},

when t ∈ R.

For this purpose, for all n ≥ 2 and any t ∈ R, choose an integer number k = k(t) by the formula

k = [n/(2 + |t|^{p+ε})] + 1,  (18)

where the symbol [z] denotes the integer part of z, and the fixed numbers p and ε are such that 1 ≤ p < 2, 0 < ε < 2 − p. It is obvious that 1 ≤ k ≤ n − 1, and

\frac{1}{2 + |t|^{p+ε}} < \frac{k}{n} ≤ \frac{1}{2 + |t|^{p+ε}} + \frac{1}{n}  (19)

for all n ≥ 2 and t ∈ R. Moreover, one can see that in the domain |t| ≤ \sqrt{n}/a

\frac{1}{2 + |t|^{p+ε}} < \frac{k}{n} ≤ \frac{3}{2 + |t|^{p+ε}}  (20)

for all n ≥ 2.

With k so chosen, represent v_{n} as a sum

v_{n} = v_{nk} + v̄_{nk},

where

v_{nk} = \frac{\sqrt{n}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−1} S_{nk}(g_{nc}),

S_{nk}(g_{nc}) = ∑_{i_{1}=1}^{k} ∑_{i_{2}=i_{1}+1}^{n−c+2} … ∑_{i_{c}=i_{c−1}+1}^{n} g_{nc}(X_{i_{1}}, X_{i_{2}}, …, X_{i_{c}});

v̄_{nk} = \frac{\sqrt{n}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−1} S̄_{nk}(g_{nc}),

where

S̄_{nk}(g_{nc}) = ∑_{k+1≤i_{1}<···<i_{c}≤n} g_{nc}(X_{i_{1}}, …, X_{i_{c}}).

We can also rewrite the variable w_{n} as a sum

w_{n} = w_{nk} + w̄_{nk},

where

w_{nk} = \frac{\sqrt{n}}{σ_{n}m} ∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d}\binom{n}{d}^{−1} S_{nk}(g_{ncd}),

S_{nk}(g_{ncd}) = ∑_{i_{1}=1}^{k} ∑_{i_{2}=i_{1}+1}^{n−d+2} … ∑_{i_{d}=i_{d−1}+1}^{n} g_{ncd}(X_{i_{1}}, X_{i_{2}}, …, X_{i_{d}});

w̄_{nk} = \frac{\sqrt{n}}{σ_{n}m} ∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d}\binom{n}{d}^{−1} S̄_{nk}(g_{ncd}),
S̄_{nk}(g_{ncd}) = ∑_{k+1≤i_{1}<···<i_{d}≤n} g_{ncd}(X_{i_{1}}, …, X_{i_{d}}).

Now we can examine the estimation of the characteristic function φ_{n}(t), t ∈ R.

Estimation of the influence of w_{nk}. By the inequality |e^{ix} − 1| ≤ |x|, x ∈ R,

φ_{n}(t) = E exp{itS_{n} + itv_{n} + itw̄_{nk}} + R_{n1}(t),  (21)

where

|R_{n1}(t)| ≤ |t| E|w_{nk}|,

and also

E|w_{nk}| ≤ \frac{\sqrt{n}}{σ_{n}m} · ∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d}\binom{n}{d}^{−1} E|S_{nk}(g_{ncd})|.

On the right-hand side

E|S_{nk}(g_{ncd})| ≤ \{\binom{n}{d} − \binom{n − k}{d}\} E|g_{ncd}|,

where

\binom{n}{d} − \binom{n − k}{d} ≤ \frac{n^{d}}{d!} · \frac{k}{n}  (22)

According to formula (9)

g_{ncd}(x_{1}, …, x_{d}) = ∑_{s=0}^{d}(−1)^{d−s} ∑_{1≤j_{1}<...<j_{s}≤d} g_{cs}^{n}(x_{j_{1}}, …, x_{j_{s}}),

where

g_{cs}^{n}(x_{1}, …, x_{s}) = Eg_{c}(x_{1}, …, x_{s}, X_{s+1}, …, X_{c}) I(|g_{c}| > n^{(2c−1)/2});

therefore,

E|g_{ncd}| ≤ 2^{d} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2}).

Moreover,

∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d}\binom{n}{d}^{−1} \frac{n^{d}}{d!} 2^{d} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})

≤ ∑_{c=3}^{m} \binom{m}{c} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2}) ∑_{d=2}^{c−1} 2^{d}\binom{c}{d}\binom{n}{d}^{−1} · \frac{n^{d}}{d!}

≤ (2m − 1)^{m} · ∑_{c=3}^{m} \binom{m}{c} E|g_{c}| · I(|g_{c}| > n^{(2c−1)/2});

here we use the fact that for n ≥ d

\binom{n}{d}^{−1} n^{d} ≤ d^{d} ≤ (c − 1)^{d}.

Thus, in (21) for all t ∈ R

|R_{n1}(t)| ≤ \frac{(2m − 1)^{m}}{σ_{n}m} · β_{n} · |t| · \frac{k}{n}.  (23)

Estimation of influence of v_{nk}. According to (21), we first write

E exp{itS_{n} + itv_{n} + itw̄_{nk}} = E exp{itS_{n} + itv̄_{nk} + itw̄_{nk}}

+it E exp{itS_{n} + itv̄_{nk} + itw̄_{nk}} v_{nk} + R_{n2}(t),  (24)

where, by the inequality |e^{ix} − 1 − ix| ≤ 2|x|^{p} for 1 ≤ p ≤ 2, x ∈ R,

|R_{n2}(t)| ≤ 2 |t|^{p} E|v_{nk}|^{p}.

Taking into account the inequality |a_{1} + ⋯ + a_{m}|^{p} ≤ m^{p−1} (|a_{1}|^{p} + ⋯ + |a_{m}|^{p}), a_{i} ∈ R, we can write the following estimate

E|v_{nk}|^{p} ≤ (m − 1)^{p−1} · m^{−p}σ_{n}^{−p}n^{p/2} ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−p} E|S_{nk}(g_{nc})|^{p},

where, according to the moment inequality for U-statistics (see, Borovskikh(1996)),

E|S_{nk}(g_{nc})|^{p} ≤ 2^{c(2−p)} E|g_{nc}|^{p} \{\binom{n}{c} − \binom{n − k}{c}\}.

Note that

g_{nc}(x_{1}, …, x_{c}) = ∑_{d=0}^{c}(−1)^{c−d} ∑_{1≤j_{1}<···<j_{d}≤c} g_{d}^{n}(x_{j_{1}}, …, x_{j_{d}}),  (25)

where

g_{d}^{n}(x_{1}, …, x_{d}) = Eg_{c}(x_{1}, …, x_{d}, X_{d+1}, …, X_{c}) I(|g_{c}| ≤ n^{(2c−1)/2}),

and moreover, for all c ≤ m ≤ n,

\binom{n}{c}^{−p} = \frac{(c!)^{p}}{(n(n − 1) ⋯ (n − c + 1))^{p}} ≤ (c!)^{p} · n^{−cp} · m^{cp} · m^{−[c]p},
\binom{m}{c}^{p} = m^{[c]p} (c!)^{−p}.

We have from (25)

E|g_{nc}|^{p} ≤ (c + 1)^{p−1} ∑_{d=0}^{c} \binom{c}{d}^{p} E|g_{d}^{n}|^{p}
≤ (c + 1)^{p−1}(∑_{d=0}^{c} \binom{c}{d}^{p}) E|g_{c}|^{p} I(|g_{c}| ≤ n^{(2c−1)/2}),  (26)

where

(c + 1)^{p−1} ∑_{d=0}^{c} \binom{c}{d}^{p} ≤ ((c + 1)!)^{p−1} · 2^{c}.

Thus, for all t ∈ R we get

|R_{n2}(t)| ≤ 3 · 2^{2m} m^{mp}((m + 1)!)^{p−1} σ_{n}^{−p} · γ_{n}(p) · |t|^{p} · \frac{k}{n}.  (27)

Then we consider the second term on the right-hand side of (24). Taking into account the definition of v_{nk} and the degeneracy property of the function g_{nc}(x_{1}, …, x_{c}), write the following representation

E exp{itS_{n} + itv̄_{nk} + itw̄_{nk}} v_{nk}

= \frac{\sqrt{n}}{σ_{n}m} φ^{k−1} (\frac{t}{σ_{n}\sqrt{n}}) · ∑_{i_{1}=1}^{k} E\{e^{itξ_{i_{1}}+it(S̄_{nk}+v̄_{nk}+w̄_{nk})}Θ_{i_{1}}^{n}\} = ζ_{1}(t) + ζ_{2}(t) + ζ_{3}(t),

where

ζ_{1}(t) = \frac{\sqrt{n}}{σ_{n}m} φ^{k−1}(\frac{t}{σ_{n}m}) ∑_{i_{1}=1}^{k} E\{(e^{itξ_{i_{1}}} − 1)(e^{itv̄_{nk}} − 1) e^{it(S̄_{nk}+w̄_{nk})} Θ_{i_{1}}^{n}\},
ζ_{2}(t) = \frac{\sqrt{n}}{σ_{n}m} φ^{k−1}(\frac{t}{σ_{n}\sqrt{n}}) ∑_{i_{1}=1}^{k} E\{(e^{itξ_{i_{1}}} − 1)(e^{itw̄_{nk}} − 1) e^{itS̄_{nk}} Θ_{i_{1}}^{n}\},
ζ_{3}(t) = \frac{\sqrt{n}}{σ_{n}m} φ^{k−1}(\frac{t}{σ_{n}\sqrt{n}}) ∑_{i_{1}=1}^{k} E\{(e^{itξ_{i_{1}}} − 1) e^{itS̄_{nk}} Θ_{i_{1}}^{n}\};

now,

S̄_{nk} = ξ_{k+1} + ⋯ + ξ_{n},

Θ_{i_{1}}^{n} = ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−1} ∑_{i_{2}=i_{1}+1}^{n−c+2} … ∑_{i_{c}=i_{c−1}+1}^{n} g_{nc}(X_{i_{1}}, X_{i_{2}}, …, X_{i_{c}}).

Estimation of ζ_{1}(t). First we write

|ζ_{1}(t)| ≤ \frac{\sqrt{n}}{σ_{n}m}|φ(\frac{t}{σ_{n}\sqrt{n}})|^{k−1} ∑_{i_{1}=1}^{k} E|e^{itξ_{1}} − 1| |e^{itv̄_{nk}} − 1| |Θ_{i_{1}}^{n}|.

By the Hölder inequality for 1 < p ≤ 2

E|e^{itξ_{1}} − 1| |e^{itv̄_{nk}} − 1| |Θ_{i_{1}}^{n}|

≤ (E|e^{itξ_{1}} − 1|^{q} |e^{itv̄_{nk}} − 1|^{q})^{1/q} (E|Θ_{i_{1}}^{n}|^{p})^{1/p},

where q = p/(p − 1). Applying inequalities |e^{ix} − 1| ≤ |x| and |e^{ix} − 1| ≤ 2|x|^{p−1}, x ∈ R, 1 ≤ p ≤ 2, and taking into account the independence of ξ_{1} and v_{nk}, we get

(E|e^{itξ_{1}} − 1|^{q} |e^{itv̄_{nk}} − 1|^{q})^{1/q} = (E|e^{itξ_{1}} − 1|^{q} · E|e^{itv̄_{nk}} − 1|^{q})^{1/q}

≤ 2 |t|^{p} (E|ξ_{1}|^{q})^{1/q} (E|v̄_{nk}|^{p})^{1−1/p},

where

(E|ξ_{1}|^{q})^{1/q} = \frac{1}{σ_{n}\sqrt{n}} (E|Z_{1}|^{q})^{1/q},

Z_{1} = g_{n1}(X_{1}) + ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} h_{nc}(X_{1}).

Then we have

E|v̄_{nk}|^{p} ≤ \frac{(m − 1)^{p−1}}{m^{p}} σ_{n}^{p} n^{p/2} ∑_{c=2}^{m} \binom{m}{c}^{p}\binom{n}{c}^{−p} E|S̄_{nk}(g_{nc})|^{p}.

Then, in the same way as we deduced (27), we get

E|S̄_{nk}(g_{nc})|^{p} ≤ 2^{(2−p)c} ∑_{k+1≤i_{1}<···<i_{c}≤n} E|g_{nc}(X_{i_{1}}, …, X_{i_{c}})|^{p}

≤ 2^{(2−p)c}\binom{n}{c} E|g_{nc}|^{p} ≤ 2^{2c}((c + 1)!)^{p−1} \binom{n}{c} E|g_{c}|^{p} I(|g_{c}| ≤ n^{(2c−1)/2}).

Therefore,

E|v̄_{nk}|^{p} ≤ \frac{(m − 1)^{p−1}}{m^{p}} σ_{n}^{−p} · n^{p/2}

× ∑_{c=2}^{m} \binom{m}{c}^{p}\binom{n}{c}^{−p+1} 2^{2c} · ((c + 1)!)^{p−1} · E|g_{c}|^{p} · I(|g_{c}| ≤ n^{(2c−1)/2}).

Here

\binom{n}{c}^{−p+1} = \frac{(c!)^{p−1}}{(n(n − 1) ⋯ (n − c + 1))^{p−1}} ≤ \frac{(c!)^{p−1}}{n^{c(p−1)}} · \frac{m^{c(p−1)}}{(m^{[c]})^{p−1}} ,

\binom{m}{c}^{p} = \binom{m}{c} · \binom{m}{c}^{p−1} = \binom{m}{c} · \frac{(m^{[c]})^{p−1}}{(c!)^{p−1}} .

Thus,

E|v̄_{nk}|^{p} ≤ \frac{(m − 1)^{p−1}}{m^{p}} σ_{n}^{−p}

× ∑_{c=2}^{m} \binom{m}{c} m^{c(p−1)} 2^{2c} · ((c + 1)!)^{p−1} n^{(2c−p(2c−1))/2} E|g_{c}|^{p} · I(|g_{c}| ≤ n^{(2c−1)/2}).

Then, we can write the estimates for all 1 ≤ i_{1} ≤ k

E|Θ_{i_{1}}^{n}|^{p} ≤ (m − 1)^{p−1} ∑_{c=2}^{m} \binom{m}{c}^{p}\binom{n}{c}^{−p}

× E| ∑_{i_{1}<i_{2}<···<i_{c}≤n} g_{nc}(X_{1}, X_{i_{2}}, …, X_{i_{c}})|^{p}

≤ (m − 1)^{p−1} · ∑_{c=2}^{m} \binom{m}{c}^{p}\binom{n}{c}^{−p} 2^{(2−p)(c−1)}

× ∑_{i_{1}<i_{2}<···<i_{c}≤n} E|g_{nc}(X_{1}, X_{i_{2}}, …, X_{i_{c}})|^{p}

≤ (m − 1)^{p−1} · ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−p}\binom{n}{c − 1} E|g_{nc}|^{p}
≤ (m − 1)^{p−1} ∑_{c=2}^{m} \binom{m}{c}^{p}\binom{n}{c}^{−p+1} · \frac{c}{n} · 2^{c+1+(2−p)(c−1)}

× ((c + 1)!)^{p−1} E|g_{c}|^{p} I(|g_{c}| ≤ n^{(2c−1)/2}).

Thus, for all 1 ≤ i_{1} ≤ k

E|Θ_{i_{1}}^{n}|^{p} ≤ (m − 1)^{p−1} ∑_{c=2}^{m} \binom{m}{c} m^{c(p−1)} c 2^{2c} ((c + 1)!)^{p−1} · n^{(2c−p(2c−1))/2}

×n^{−1−p/2} E|g_{c}| I(|g_{c}| ≤ n^{(2c−1)/2}).

Therefore, we have shown that for t ∈ R, for all 1 < p ≤ 2, and q = p/(p − 1) the following holds

|ζ_{1}(t)| ≤ 2^{2m} · m^{mp} ((m + 1)!)^{p−1} σ_{n}^{−p−1} (E|Z_{1}|^{q})^{1/q} |t|^{p}
× |φ(\frac{t}{σ_{n}\sqrt{n}})|^{k−1} n^{−(2−p)/(2p)} γ_{n}(p).  (28)

Estimation of ζ_{2}(t). First we write the inequality

|ζ_{2}(t)| ≤ \frac{\sqrt{n}}{σ_{n}m} |φ(\frac{t}{σ_{n}\sqrt{n}})|^{k−1} ∑_{i_{1}=1}^{k} E|e^{itξ_{1}} − 1| |e^{itw̄_{nk}} − 1| |Θ_{i_{1}}^{n}|.

For 1 < p ≤ 2, by the Hölder inequality and taking into account inequalities |e^{ix} − 1| ≤ 2 |x|^{(p−1)/p}, we have

E|e^{itξ_{1}} − 1| |e^{itw̄_{nk}} − 1| |Θ_{i_{1}}^{n}|

≤ (E|e^{itξ_{1}} − 1|^{q})^{1/q} · (E|e^{itw̄_{nk}} − 1|^{q})^{1/q} (E|Θ_{i_{1}}^{n}|^{p})^{1/p}

≤ \frac{2 · |t|^{2−1/p}}{σ_{n}\sqrt{n}} (E|Z_{1}|^{q})^{1/q} (E|w̄_{nk}|)^{1−1/p} (E|Θ_{i_{1}}^{n}|^{p})^{1/p}.

Here

E|w̄_{nk}| ≤ \frac{\sqrt{n}}{σ_{n}m} ∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d}\binom{n}{d}^{−1} E|S̄_{nk}(g_{ncd})|,

and moreover,

E|S̄_{nk}(g_{ncd})| ≤ ∑_{k+1≤i_{1}<...<i_{d}≤n} E|g_{ncd}| ≤ \binom{n}{d} E|g_{ncd}|

≤ 2^{d}\binom{n}{d} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2}),

∑_{d=2}^{m−1} ∑_{c=d+1}^{m} \binom{m}{c}\binom{c}{d} 2^{d} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})

= ∑_{c=3}^{m} \binom{m}{c} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2}) · (∑_{d=2}^{c−1} \binom{c}{d} 2^{d})
≤ ∑_{c=3}^{m} \binom{m}{c} 3^{c} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2}).

Therefore,

E|w̄_{nk}| ≤ \frac{1}{σ_{n}m} ∑_{c=3}^{m} \binom{m}{c} 3^{c} \{\sqrt{n} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\}.

For t ∈ R and all 1 < p ≤ 2 and q = p/(p − 1) this leads to the inequality

|ζ_{2}(t)| ≤ 2^{2m/p−1} 3^{m(1−1/p)} · m^{m−1} ((m + 1)!)^{1−1/p}

× σ_{n}^{−3+1/p} · (E|Z_{1}|^{q})^{1/q} |t|^{2−1/p} · |φ(\frac{t}{σ_{n}\sqrt{n}})|^{k−1} n^{−(2−p)/p} · γ_{n}^{1/p}(p) · β_{n}^{1−1/p}. (29)

Estimation of ζ_{3}(t). According to the definition, and taking into account the independence of X_{1}, …, X_{n}, and the degeneracy of the functions g_{nc}(x_{1}, …, x_{c}), c = 2,…, m, we can write

ζ_{3}(t) = \frac{\sqrt{n}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−1} φ^{n−c}(\frac{t}{σ_{n}\sqrt{n}})

× ∑_{i_{1}=1}^{k} ∑_{i_{2}=i_{1}+1}^{n−c+2} … ∑_{i_{c}=i_{c−1}+1}^{n} E{(e^{itξ_{1}} − 1) e^{itξ_{2}+...+itξ_{c}} g_{nc}(X_{1}, X_{2}, …, X_{c})}

= \frac{\sqrt{n}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−1}[\binom{n}{c} − \binom{n − k}{c}] φ^{n−c}(\frac{t}{σ_{n}\sqrt{n}})

× E{g_{nc}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{itξ_{s}} − 1)}.  (30)

From (28)–(30) we derive the necessary estimate in (24).

Estimation of the influence of w̄_{nk}. Now consider the first expression on the right-hand side of (24). First, we can write

E exp\{itS_{n} + itv̄_{nk} + itw̄_{nk}\} = E exp\{itS_{n} + itv̄_{nk}\} + φ^{k}(\frac{t}{σ_{n}\sqrt{n}}) R_{n3}(t), (31)

then, using the inequality |e^{ix} − 1| ≤ |x|, x ∈ R, we have

|R_{n3}(t)| ≤ |t| · E|w̄_{nk}|,

i.e., for all t ∈ R,

|R_{n3}(t)| ≤ \frac{|t|}{σ_{n}m} 3^{m} · β_{n}.  (32)

Estimation of the influence of v̄_{nk}. According to the definition in (31),

E exp\{itS_{n} + itv̄_{nk}\} = E exp\{itS_{n}\} + it Ee^{itS_{n}} v̄_{nk} + φ^{k}(\frac{t}{σ_{n}\sqrt{n}}) R_{n4}(t), (33)

where for 1 ≤ p ≤ 2,

|R_{n4}(t)| ≤ 2 |t|^{p} E|v̄_{nk}|^{p},

holds, i.e., for t ∈ R,

|R_{n4}(t)| ≤ 2^{2m+1} · m^{mp−1} ((m + 1)!)^{p−1} σ_{n}^{−p} · |t|^{p} · γ_{n}(p).  (34)

In order to estimate the second term on the right-hand side of (33), write the representation

Ee^{itS_{n}} v̄_{nk} = \frac{\sqrt{n}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−1} ∑_{k+1≤i_{1}<···<i_{c}≤n} E\{e^{itS_{n}} g_{nc}(X_{i_{1}}, …, X_{i_{c}})\}

= \frac{\sqrt{n}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c}\binom{n}{c}^{−1}\binom{n − k}{c} φ^{n−c}(\frac{t}{σ_{n}\sqrt{n}})

×E{g_{nc}(X_{1}, …, X_{c}) · ∏_{s=1}^{c}(e^{itξ_{s}} − 1)}.  (35)

Write

ζ_{4}(t) = ζ_{3}(t) + Ee^{itS_{n}} v̄_{nk}.

Then one can see from (30) and (35), that

ζ_{4}(t) = \frac{\sqrt{n}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c} φ^{n−c}(\frac{t}{σ_{n}\sqrt{n}}) E\{g_{nc}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{itξ_{s}} − 1)\}.  (36)

First, by the degeneracy property of the function g_{nc}(x_{1}, …, x_{c}), we can write

E{g_{nc}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{itξ_{s}} − 1)} = E{g_{nc}(X_{1}, …, X_{c}) · e^{it ∑_{s=1}^{c} ξ_{s}}}

= e^{− \frac{itc}{σn\sqrt{n}} (Eg_{1}^{n}(X_{1})+Eh̄^{n}(X_{1}))} E\{g_{nc}(X_{1}, …, X_{c}) e^{\frac{it}{σn\sqrt{n}} ∑_{s=1}^{c}(g_{1}^{n}(X_{s})+h̄^{n}(X_{s}))}\}

= e^{− \frac{itc}{σn\sqrt{n}} (Eg_{1}^{n}(X_{1})+Eh̄^{n}(X_{1}))} E\{g_{nc}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{\frac{it}{σn\sqrt{n}} (g_{1}^{n}(X_{s})+h̄^{n}(X_{s}))} − 1)\}, (37)

where h̄^{n}(X_{s}) = ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} h_{c}^{n}(X_{s}).

According to (25),

g_{nc}(x_{1}, …, x_{c}) = g_{c}^{n}(x_{1}, …, x_{c}) + ∑_{d=0}^{c−1}(−1)^{c−d} ∑_{1≤j_{1}<...<j_{d}≤c} g_{d}^{n}(x_{j_{1}}, …, x_{j_{d}}),

where for all d = 0, 1,…, c − 1, we have the formula

g_{d}^{n}(x_{1}, …, x_{d}) = −Eg_{c}(x_{1}, …, x_{d}, X_{d+1}, …, X_{c}) I(|g_{c}| > n^{(2c−1)/2}).

Therefore, in (37), we can write

E{g_{nc}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{itξ_{s}} − 1)}

= E\{g_{c}^{n}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{\frac{it}{σn\sqrt{n}} (g_{1}^{n}(X_{s})+h̄^{n}(X_{s}))} − 1)\} + R_{n5}(t),  (38)

where

|R_{n5}(t)| ≤ 2^{c} · E|g_{c}| I(|g_{c}| > n^{(2c−1)/2}).

Using the inequality |e^{ix} − 1| ≤ |x|, x ∈ R, and the well-known formula

∏_{s=1}^{c}(a_{s} + b_{s}) = ∑_{d=0}^{c} ∑_{1≤j_{1}<···<j_{d}≤c} ∏_{s=1}^{d} b_{j_{s}} ∏_{s=d+1}^{c} a_{j′_{s}}, a_{s}, b_{s} ∈ R,

where {j′_{d+1}, …, j′_{c}} = {1, …, c}\{j_{1}, …, j_{d}}, the following relations can be obtained

|E\{g_{c}^{n}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{\frac{it}{σn\sqrt{n}} (g_{1}^{n}(X_{s})+h̄^{n}(X_{s}))} − 1)\}|

≤ |t|^{c} · n^{−c/2} · σ_{n}^{−c} E|g_{c}^{n}(X_{1}, …, X_{c})| ∏_{s=1}^{c}(|g_{1}^{n}(X_{s})| + |h̄^{n}(X_{s})|)
= |t|^{c} · n^{−c/2} σ_{n}^{−c} ∑_{d=0}^{c} ∑_{1≤j_{1}<...<j_{d}≤c} E|g_{c}^{n}(X_{1}, …, X_{c})|

× ∏_{s=1}^{d} |h̄^{n}(X_{j_{s}})| · ∏_{s=d+1}^{c} |g_{1}^{n}(X_{j′_{s}})|

= |t|^{c} n^{−c/2} σ_{n}^{−c} ∑_{d=0}^{c} \binom{c}{d} E|g_{c}^{n}| ∏_{s=1}^{d} |h̄^{n}(X_{s})| · ∏_{s=d+1}^{c} |g_{1}^{n}(X_{s})|
= |t|^{c} · n^{−c/2} σ_{n}^{−c} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|

+ |t|^{c} · n^{−c/2} σ_{n}^{−c} ∑_{d=1}^{c} \binom{c}{d} E|g_{c}^{n}| ∏_{s=1}^{d} |h̄^{n}(X_{s})| ∏_{s=d+1}^{c} |g_{1}^{n}(X_{s})|. (39)

Therefore,

|ζ_{4}(t)| ≤ \frac{1}{σ_{n}m} · ∑_{c=2}^{m} \binom{m}{c} 2^{c} |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c} · \{\sqrt{n} E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\}

+\frac{1}{σ_{n}m} · ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} |t|^{c} |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c} · n^{− \frac{c}{2} + \frac{1}{2}} · E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|

+\frac{1}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} |t|^{c} |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c} n^{− \frac{c}{2} + \frac{1}{2}}

× ∑_{d=1}^{c} \binom{c}{d} E|g_{c}^{n}| ∏_{s=1}^{d} |h̄^{n}(X_{s})| ∏_{s=d+1}^{c} |g_{1}^{n}(X_{s})|.  (40)

If 2 ≤ c ≤ m, and n ≥ m, then

|φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c} ≤ |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−m}.

Since

∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{1−2/c} ≤ n^{c/2−1},

then

n^{− \frac{c}{2} + \frac{1}{2}} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})| ≤ E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}.

By the Hölder inequality, and taking into account that Eg_{1}^{2} = 1,

E|g_{c}^{n}| ∏_{s=1}^{d} |h̄^{n}(X_{s})| ∏_{s=d+1}^{c} |g_{1}^{n}(X_{s})| ≤ (E|g_{c}^{n}|^{2})^{1/2}(E|h̄^{n}(X_{1})|^{2})^{1/2},

and therefore, we can write the following inequality

∑_{d=1}^{c} \binom{c}{d} E|g_{c}^{n}| ∏_{s=1}^{d} |h̄^{n}(X_{s})| ∏_{s=d+1}^{c} |g_{1}^{n}(X_{s})|

≤ c (E|h̄^{n}(X_{1})|^{2})^{1/2}(1 + (E|h̄^{n}(X_{1})|^{2})^{1/2})^{c−1}(E|g_{c}^{n}|^{2})^{1/2}.

Therefore, inequality (40) takes the form

|ζ_{4}(t)| ≤ \frac{2^{m}}{σ_{n}m} · |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−m} · β_{n}

+ \frac{1}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} |t|^{c} |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}

+ \frac{1}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} |t|^{c} |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c} · c · (1 + (E|h̄^{n}(X_{1})|^{2})^{1/2})^{c−1}

× (E|h̄^{n}(X_{1})|^{2})^{1/2} · (n^{−c+1} E|g_{c}^{n}|^{2})^{1/2}.  (41)

Note that in (41)

E|h̄^{n}(X_{1})|^{2} ≤ 2^{m} · β_{n}.  (42)

In the following two cases may occur, β_{n} ≥ 1, or β_{n} ≤ 1. If β_{n} ≥ 1, then always Δ_{n} ≤ 1 ≤ β_{n}. This implies that Δ_{n} ≤ c_{1}β_{n} + c_{2} for any c_{1} ≥ 1 and c_{2} ≥ 0. This proves inequality (3) in the case β_{n} ≥ 1. Below, we consider the case β_{n} ≤ 1 starting with (41). Using (42), we have from (41)

|ζ_{4}(t)| ≤ \frac{2^{m}}{σ_{n}m} · |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−m} · β_{n}

+ \frac{1}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} |t|^{c}|φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}

+ \frac{2^{m/2} · β_{n}^{1/2}}{σ_{n}m} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} |t|^{c} · |φ(\frac{t}{σ_{n}\sqrt{n}})|^{n−c}

× c · (1 + 2^{m/2})^{c−1} · (n^{−c+1} E|g_{c}^{n}|^{2})^{1/2}.  (43)

Thus, we have proved that for t ∈ R,

φ_{n}(t) − φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) = R_{n1}(t) + R_{n2}(t) + it (ζ_{1}(t) + ζ_{2}(t))

+it φ^{k}(\frac{t}{σ_{n}\sqrt{n}}) (R_{n3}(t) + R_{n4}(t)) + it ζ_{4}(t),  (44)

where each term on the right-hand side is already estimated in (23), (27)–(29), (32), (34), and (43), respectively. Further, let 1 ≤ p < 2 in (27), and let p = 2 in (28), (29), and (34). Then, using (44), and taking into account (14), we obtain

∫_{−\sqrt{n}/a}^{\sqrt{n}/a}|φ_{n}(t) − φ^{n}(\frac{t}{σ_{n}\sqrt{n}})| |t|^{−1} dt

≤ ∫_{−\sqrt{n}/a}^{\sqrt{n}/a}(|R_{n1}(t)| + |R_{n2}(t)|) |t|^{−1} dt + ∫_{−\sqrt{n}/a}^{\sqrt{n}/a}(|ζ_{1}(t)| + |ζ_{2}(t)|) dt
+ ∫_{−\sqrt{n}/a}^{\sqrt{n}/a}(|R_{n3}(t)| + |R_{n4}(t)|) e^{− \frac{t^{2}}{3} · \frac{k}{n}} dt + ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} |ζ_{4}(t)| dt.  (45)

Simple calculations show that

∫_{−\sqrt{n}/a}^{\sqrt{n}/a}(|R_{n1}(t)| + |R_{n2}(t)|) |t|^{−1} dt

≤ \frac{3(2m − 1)^{m}}{σ_{n}m} β_{n} ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} \frac{dt}{2 + |t|^{p+ε}}

+ 9 · 2^{2m} m^{mp} ((m + 1)!)^{p−1} σ_{n}^{−p} γ_{n}(p) ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} \frac{|t|^{p−1} dt}{2 + |t|^{p+ε}}

≤ \frac{9(2m − 1)}{σ_{n}m} (p + ε − 1)^{−1} β_{n}

+ 36 · 2^{2m} m^{mp} ((m + 1)!)^{p−1} ε^{−1} σ_{n}^{−p} · γ_{n}(p),

where 1 ≤ p < 2, and 0 < ε < 2 − p.

In order to estimate other integrals in (45), we shall use the following inequality

e^{− \frac{t^{2}}{3n} (k−1)} ≤ e · e^{− \frac{1}{3} |t|^{2−p−ε}},

which holds for all |t| ≤ \sqrt{n}/a, and k from (18). Write the following estimates

∫_{−\sqrt{n}/a}^{\sqrt{n}/a}(|ζ_{1}(t)| + |ζ_{2}(t)|) dt ≤ 2^{2m+1} m^{2m} (m + 1)! σ_{n}^{−3} ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} dt · t^{2} e^{− \frac{t^{2}}{3n} (k−1)}

+2^{m+1} m^{m−1} ((m + 1)!)^{1/2}3^{m/2} σ_{n}^{−5/2} β_{n}^{1/2} γ_{n}^{1/2}(2) ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} dt · |t|^{3/2} e^{− \frac{t^{2}}{3n} (k−1)}

≤ \frac{e}{(2 − p − ε)} · 3^{\frac{3}{2−p−ε}} · Γ(\frac{3}{2 − p − ε}) 2^{2m+2} m^{2m} (m + 1)! σ_{n}^{−3} · γ_{n}(2)

+ \frac{e}{(2 − p − ε)} · 3^{\frac{m}{2} + \frac{5}{2(2−p−ε)}} · Γ(\frac{5}{2 (2 − p − ε)})

× 2^{m+2} m^{m−1} ((m + 1)!)^{1/2} σ_{n}^{−5/2} · β_{n}^{1/2} γ_{n}^{1/2}(2),

where Γ(z) is the gamma function;

∫_{−\sqrt{n}/a}^{\sqrt{n}/a} dt (|R_{n3}(t)| + |R_{n4}(t)|) |φ(\frac{t}{σ_{n}\sqrt{n}})|^{k} ≤ \frac{3^{m} e}{σ_{n}m} β_{n} ∫_{−∞}^{∞} dt |t| e^{− \frac{1}{3} |t|^{2−p−ε}}

+2^{2m+1} m^{2m−1} (m + 1)! e σ_{n}^{−2} γ_{n}(2) ∫_{−∞}^{∞} dt t^{2} e^{− \frac{1}{3} |t|^{2−p−ε}}

≤ \frac{2 e 3^{m}}{σ_{n}m} 3^{\frac{2}{2−p−ε}} Γ(\frac{2}{2 − p − ε}) β_{n}

+\frac{2^{2m+2} m^{2m} (m + 1)! e}{(2 − p − ε)} 3^{\frac{3}{2−p−ε}} · Γ(\frac{3}{2 − p − ε}) σ_{n}^{−2} · γ_{n}(2);

∫_{−\sqrt{n}/a}^{\sqrt{n}/a} |ζ_{4}(t)| dt ≤ \frac{1}{σ_{n}m} 2^{m} e^{m/3} β_{n} ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} dt e^{− \frac{t^{2}}{3}}

+\frac{1}{σ_{n}m} e^{m/3} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} \{∫_{−\sqrt{n}/a}^{\sqrt{n}/a} dt |t|^{c} e^{− \frac{t^{2}}{3}} \}E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}

+\frac{2^{m/2} e^{m/3}}{σ_{n}m} ·β_{n}^{1/2} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} \{∫_{−\sqrt{n}/a}^{\sqrt{n}/a} dt |t|^{c} e^{− \frac{t^{2}}{3}} \}c·(1+2^{m/2})^{c−1} (n^{−c+1} E|g_{c}^{n}|^{2})^{1/2}

≤ \frac{1}{σ_{n}m} 2^{m} e^{m/3} 3\sqrt{3π} β_{n} + \frac{1}{σ_{n}m} e^{m/3} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} 3^{(c+1)/2} Γ(c + \frac{1}{2})

×E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}} + \frac{1}{σ_{n}m} 2^{m/2} e^{m/3} β_{n}^{1/2} ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} 3^{(c+1)/2} Γ(c + \frac{1}{2})

× c (1 + 2^{m/2})^{c−1} · (n^{−c+1} E|g_{c}^{n}|^{2})^{1/2} ≤ \frac{3\sqrt{3π}}{σ_{n}m} · 2^{m} e^{m/3} β_{n}

+\frac{\sqrt{π}}{σ_{n}m} e^{m/3} 3^{(m+1)/2} m! ∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}

+σ_{n}^{−1} \sqrt{π} m! 6^{m/2} e^{m/3}(1 + σ_{n}^{−2} (1 + 2^{m/2})^{2})^{m/2} β_{n}^{1/2} γ_{n}^{1/2}(2),

since 2 ≤ c ≤ m, Γ(c + \frac{1}{2}) ≤ \sqrt{π} c!, and

∑_{c=2}^{m} \binom{m}{c} σ_{n}^{−c} 3^{(c+1)/2} Γ(c + \frac{1}{2}) c (1 + 2^{m/2})^{c−1} (n^{−c+1} E|g_{c}^{n}|^{2})^{1/2}

≤ \sqrt{π} m! m 3^{m/2} (1 + σ_{n}^{−2}(1 + 2^{m/2})^{2})^{m/2} γ_{n}^{1/2}(2).

Thus, the estimation of all the integrals on the right-hand side of (45), leads to the inequality

∫_{−\sqrt{n}/a}^{\sqrt{n}/a}|φ_{n}(t) − φ^{n}(\frac{t}{σ_{n}\sqrt{n}})| |t|^{−1} dt ≤ β_{n}\{\frac{9 (2m − 1)}{σ_{n}m} (p + ε − 1)^{−1}

+ \frac{1}{σ_{n}m} 2^{m} e^{m/3} 3\sqrt{3π} + \frac{2 e · 3^{m}}{σ_{n}m} 3^{\frac{2}{2−p−ε}} Γ(\frac{2}{2 − p − ε})\}

+ γ_{n}(p) {36 · 2^{2m} m^{mp} ((m + 1)!)^{p−1} · ε^{−1} · σ_{n}^{−p}}
+ γ_{n}(2) \{\frac{e}{(2 − p − ε)} 3^{\frac{3}{2−p−ε}} Γ(\frac{3}{2 − p − ε}) 2^{2m+2} m^{2m} (m + 1)!

+ \frac{e}{(2 − p − ε)} · 2^{2m+2} m^{2m} (m + 1)! 3^{\frac{3}{2−p−ε}} Γ(\frac{3}{2 − p − ε}) · σ_{n}^{−2}\}

+ β_{n}^{1/2} · γ_{n}^{1/2}(2) \{σ_{n}^{−5/2} \frac{e}{(2 − p − ε)} · 3^{\frac{m}{2} + \frac{5}{2(2−p−ε)}}

× Γ(\frac{5}{2(2 − p − ε)}) · 2^{m+2} m^{m−1} ((m + 1)!)^{1/2}

+ σ_{n}^{−1}\sqrt{π} 6^{m/2} e^{m/3} · m! (1 + σ_{n}^{−2}(1 + 2^{m/2})^{2})^{m/2}\}

+ \frac{\sqrt{π}}{σ_{n}m} 2^{m/2} e^{m/3} 3^{(m+1)/2} m! ∑_{c=2}^{m} \binom{m}{c} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}.  (46)

The estimation of σ_{n}^{2}. According to the definition

σ_{n}^{2} = Eg_{n1}^{2}(X_{1}) + 2 ∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} Eg_{n1}(X_{1})h_{nc}(X_{1}) + E(∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} h_{nc}(X_{1}))^{2}.

Here

Eg_{n1}^{2} = 1 − Eg_{1}^{2} I(|g| > \sqrt{n}) − (Eg_{1} I(|g_{1}| > \sqrt{n}))^{2};

|Eg_{n1}(X_{1}) h_{nc}(X_{1})| ≤ \sqrt{n} E|h_{nc}| ≤ 2\sqrt{n} · E|g_{c}| I(|g_{c}| > n^{(2c−1)/2});

E(∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} h_{nc}(X_{1}))^{2} ≤ 2\sqrt{n} · (∑_{c=2}^{m} \binom{m}{c} \frac{c}{m}) · (∑_{c=2}^{m} \binom{m}{c} \frac{c}{m} E|h_{nc}|)

≤ 2^{m+1} · ∑_{c=2}^{m} \binom{m}{c} \{\sqrt{n} · E|g_{c}| I(|g_{c}| > n^{(2c−1)/2})\} = 2^{m+1} · β_{n}.

Thus,

σ_{n}^{2} ≥ 1 − 2 Eg_{1}^{2} I(|g_{1}| > \sqrt{n}) − 2 β_{n},
σ_{n}^{2} ≤ 1 − Eg_{1}^{2} I(|g_{1}| > \sqrt{n}) + 2 (2^{m} + 1) β_{n}.  (47)

This implies

|σ_{n}^{2} − 1| ≤ α,

where α = 2 · Eg_{1}^{2} I(|g_{1}| > \sqrt{n}) + 2 (2^{m} + 1) β_{n}. One of the two cases may occur, 2α ≤ 1, and 2α ≥ 1. If it is 2α ≥ 1, then the needed inequality (3) is obvious, since Δ_{n} ≤ 1 ≤ 2α. In the case of 2α ≤ 1 we can use (10) and (46). First note that

\frac{1}{σ_{n}^{2}} ≤ \frac{1}{1 − α} ≤ 1 + 2α ≤ 2,

and therefore, in (10)

σ_{n}^{−1} |1 − σ_{n}^{2}| ≤ 4 Eg_{1}^{2} I(|g_{1}| > \sqrt{n}) + 4 (2^{m} + 1) β_{n}.  (48)

Join estimates (10), (13), (16), (46) and (48). As a result, we get for any 1 ≤ p < 2 and 0 < ε < 2 − p the following inequality

Δ_{n} ≤ (28 · \sqrt{\frac{6e}{π}} + \frac{24 · 16}{π\sqrt{π}}) E|g_{1}^{n}(X_{1})|^{3} · \frac{1}{\sqrt{n}}

+ (1 + \frac{1}{\sqrt{2π}} + \frac{4}{\sqrt{2πe}}) Eg_{1}^{2} I(|g_{1}| > \sqrt{n})

+ (3 + \frac{1}{\sqrt{2πe}} 4(2^{m} + 1) + 7 · \sqrt{\frac{6e}{π}} · 2^{2m+5} + 18 · \sqrt{2} · (p + ε − 1)^{−1}

+ 2^{m} e^{m/3} 3\sqrt{6π} + 2\sqrt{2} e · 3^{m+2/(2−p−ε)} · Γ(\frac{2}{2 − p − ε})) · β_{n}

+ (3^{2} · 2^{2m+2+p/2} · m^{mp} ((m + 1)!)^{p−1} · ε^{−1}) · γ_{n}(p)

+ (\frac{e}{2 − p − ε} · 3^{\frac{3}{2−p−ε} +1} · Γ(\frac{3}{2 − p − ε}) · 2^{2m+2} · m^{2m} (m + 1)!) · γ_{n}(2)

+ (\frac{\sqrt{2} · e}{2 − p − ε} 3^{\frac{m}{2} + \frac{5}{2(2−p−ε)}} · Γ(\frac{5}{2(2 − p − ε)}) · 2^{m+3} · m^{m−1} ((m + 1)!)^{1/2}

+ \sqrt{2π} 6^{m/2} e^{m/3} m! (1 + 2(1 + 2^{m/2})^{2})^{m/2}) · β_{n}^{1/2} · γ_{n}^{1/2}(2)
+\sqrt{2π} 2^{m/2} e^{m/3} 3^{(m+1)/2} · m! ∑_{c=2}^{m} \binom{m}{c} E|g_{c}^{n}| ∏_{s=1}^{c} |g_{1}^{n}(X_{s})|^{2/c} · \frac{1}{\sqrt{n}}.  (49)

Note that in (49) γ_{n}(2) ≤ γ_{n}(p) and β_{n}^{1/2}γ_{n}^{1/2}(2) ≤ (β_{n} + γ_{n}(2))/2. Then, letting ε = (2 − p)/2 in (49) and simplifying some constants on the right-hand side of this inequality, we get the proof of Theorem 1.

In what follows we shall apply the inequality (3).

Proof of Corollary 1. First we estimate the value of β_{n}. Consider

\sqrt{n}E|g_{c}|I(|g_{c}| > n^{(2c−1)/2}) = n^{c}P (|g_{c}| > n^{(2c−1)/2}) + \sqrt{n} ∫_{n^{(2c−1)/2}}^{∞} P (|g_{c}| > x)dx

≤ B_{c}(n^{(2c−1)/2}, γ_{c}) + \sqrt{n}B_{c}(n^{(2c−1)/2}, γ_{c}) ∫_{n^{(2c−1)/2}}^{∞} x^{−γ_{c}}dx
= 2cB_{c}(n^{(2c−1)/2}, γ_{c}).  (50)

Therefore,

β_{n} ≤ 2 ∑_{c=2}^{m} \binom{m}{c}cB_{c}(n^{(2c−1)/2}, γ_{c}) → 0,

when n → ∞. Further, the value γ_{n}(p) has to be estimated. Let 4/3 < p ≤ 2. Denote

b_{n}(c, p) := n^{(2c−p(2c−1))/2}E|g_{c}|^{p}I(|g_{c}| ≤ n^{(2c−1)/2})  (51)

for 2 ≤ c ≤ m.

Estimate b_{n}(c, p). For this purpose consider a constant α_{c}, 0 < α_{c} < (2c−1)/2, and

∫_{0}^{n^{(2c−1)/2}} x^{p}dP (|g_{c}| ≤ x) = − ∫_{0}^{n^{(2c−1)/2}} x^{p}dP (|g_{c}| > x)

= −n^{p(2c−1)/2}P (|g_{c}| > n^{(2c−1)/2}) + ∫_{0}^{n^{(2c−1)/2}} x^{p−1}P (|g_{c}| > x)dx

≤ p ∫_{0}^{n^{(2c−1)/2}} x^{p−1}P (|g_{c}| > x)dx

= p ∫_{0}^{n^{αc}} x^{p−1}P (|g_{c}| > x)dx + p ∫_{n^{αc}}^{n^{(2c−1)/2}} x^{p−γ_{c}−1}(x^{γ_{c}}P (|g_{c}| > x))dx

≤ B_{c}(0, γ_{c})p ∫_{0}^{n^{αc}} x^{p−γ_{c}−1}dx + B_{c}(n^{α_{c}}, γ_{c})p ∫_{n^{αc}}^{n^{(2c−1)/2}} x^{p−γ_{c}−1}dx
≤ \frac{p}{p − γ_{c}} B_{c}(0, γ_{c})n^{α_{c}(p−γ_{c})} + \frac{p}{p − γ_{c}} B_{c}(n^{α_{c}}, γ_{c})n^{\frac{(2c−1)}{2} (p−γ_{c})}.

Therefore, for α_{c} = (2c − 1)/4 the following inequality is true

b_{n}(c, p) ≤ \frac{p}{p − γ_{c}} B_{c}(0, γ_{c}) · n^{− \frac{2c−1}{4} (p−γ_{c})} + \frac{p}{p − γ_{c}} B_{c}(n^{(2c−1)/4}, γ_{c}),  (52)

where 2 ≤ c ≤ m, 4/3 < p ≤ 2. Consequently,

γ_{n}(p) → 0, n → ∞.

This proves Corollary 1.

Proof of Theorem 2. Under the conditions of the theorem we estimate the expression on the right-hand side of (3). First note that

Eg_{1}^{2}I(|g_{1}| > \sqrt{n}) ≤ \frac{3}{\sqrt{n}}B_{1}(\sqrt{n}, δ_{1})

≤ \frac{3}{\sqrt{n}}B_{1}(0, δ_{1}).  (53)

Analogously to (50) it can be deduced that

\sqrt{n}E|g_{c}|I(|g_{c}| > n^{(2c−1)/2}) ≤ \frac{2c + 1}{2}B_{c}(n^{(2c−1)/2}, δ_{c})\frac{1}{\sqrt{n}}
≤ \frac{2c + 1}{2}B_{c}(0, δ_{c})\frac{1}{\sqrt{n}}.  (54)

Moreover

β_{n} ≤ \frac{2m + 1}{2} ∑_{c=2}^{m} \binom{m}{c}B_{c}(0, δ_{c})\frac{1}{\sqrt{n}}.  (55)

Consider the value of b_{n}(c, p) in the formula (51) when 5/3 < p ≤ 2 and c = 2, …, m. For this purpose it can be written analogously to (52)

E|g_{c}|^{p}I(|g_{c}| ≤ n^{(2c−1)/2}) = −n^{p(2c−1)/2}P (|g_{c}| > n^{(2c−1)/2})

+p ∫_{0}^{n^{(2c−1)/2}} x^{p−1}P (|g_{c}| > x)dx ≤ p ∫_{0}^{n^{(2c−1)/2}} x^{p−1}P (|g_{c}| > x)dx

= p ∫_{0}^{n^{(2c−1)/2}} x^{p−δ_{c}−1}(x^{δ_{c}}P (|g_{c}| > x))dx ≤ B_{c}(0, δ_{c})p ∫_{0}^{n^{(2c−1)/2}} x^{p−δ_{c}−1}dx
= \frac{p}{p − δ_{c}} B_{c}(0, δ_{c})n^{\frac{2c−1}{2} (p−δ_{c})}  (56)

for all p > δ_{c}.

Consequently, for 5/3 < p ≤ 2 and c = 2, 3, …, m

b_{n}(c, p) ≤ \frac{p}{p − 5/3}B_{c}(0, δ_{c})\frac{1}{\sqrt{n}}.  (57)

Therefore

γ_{n}(p) ≤ \frac{p}{p − 5/3} ∑_{c=2}^{m} \binom{m}{c}B_{c}(0, δ_{c})\frac{1}{\sqrt{n}}.  (58)

Put p = 15/8 in (3). Then (4) can be obtained from (53), (55) and (58). Note for 2 ≤ c ≤ m by Hölder inequality

E|g_{c}| ∏_{s=1}^{c} |g_{1}(X_{s})|^{2/c} ≤ (E|g_{c}|^{\frac{3c}{3c−2}} )^{\frac{3c−2}{3c}} (E|g_{1}|^{3})^{\frac{2}{3}}  (59)

where

E|g_{c}|^{\frac{3c}{3c−2}} ≤ 1 + \frac{3c(2c − 1)}{2(c − 1)}B_{c}(0, δ_{c}), δ_{c} = \frac{3c}{3c − 2} + \frac{2(c − 1)}{(2c − 1)(3c − 2)}.  (60)

Proof of Corollary 2. In well-known inequality

a · b ≤ p^{−1}a^{p} + q^{−1}b^{q}, a ≥ 0, b ≥ 0, p^{−1} + q^{−1} = 1

we take for c = 1, …, m

a = |g_{c}|, b = ∏_{s=1}^{c} |g_{1}(X_{s})|^{2/c}, p = δ_{c} = \frac{2c + 1}{2c − 1}, q = \frac{δ_{c}}{δ_{c} − 1} = \frac{2c + 1}{2}.

Then we have

E |g_{c}| ∏_{s=1}^{c} |g_{1}(X_{s})|^{2/c} ≤ \frac{2c − 1}{2c + 1} E |g_{c}|^{δ_{c}} + \frac{2}{2c + 1}(E |g_{c}|^{(2c+1)/c})^{c}.

Here by Hölder inequality

E |g_{1}|^{(2c+1)/c} = E |g_{1}|^{\frac{3}{c}} · |g_{1}|^{\frac{2(c−1)}{c}} ≤
≤ (E |g_{1}|^{3})^{1/c}(E |g_{1}|^{2})^{\frac{c−1}{c}} = (E |g_{1}|^{3})^{1/c}.

Therefore for c = 1, …, m

E |g_{c}| ∏_{s=1}^{c} |g_{1}(X_{s})|^{2/c} ≤ \frac{2c − 1}{2c + 1}E |g_{c}|^{δ_{c}} + \frac{2}{2c + 1} · E |g_{1}|^{3}.

This proves Corollary 2.

Proof of Theorem 3. According to (54)

β_{n} ≤ \frac{2m + 1}{2} ∑_{c=2}^{m} \binom{m}{c}B_{c}(n^{(2c−1)/2}, δ_{c})\frac{1}{\sqrt{n}}.  (61)

Estimate γ_{n}(p) for 5/3 < p ≤ 2. Analogously to the deduction of (52) and (57) the following inequality can be obtained

b_{n}(c, p) ≤ \frac{p}{p − δ_{c}} B_{c}(0, δ_{c})n^{α_{c}(p−δ_{c})+(2c−p(2c−1))/2}

+ \frac{p}{p − δ_{c}} B_{c}(n^{α_{c}}, δ_{c})\frac{1}{\sqrt{n}},

which is true for all 5/3 < p ≤ 2, c = 2, …, m, and every α_{c} such that 0 < α_{c} < (2c − 1)/2. Assume α_{c} = (2c − 1)/4. Taking into account that 1 < δ_{c} ≤ 5/3 and B_{c}(n^{α_{c}}, δ_{c}) ≤ B_{c}(\sqrt{n}, δ_{c}) the following value can be estimated

b_{n}(c, p) ≤ \frac{p}{p − 5/3}\{B_{c}(0, δ_{c})n^{− \frac{1}{4} (3p−5)} + B_{c}(\sqrt{n}, δ_{c})\}\frac{1}{\sqrt{n}},  (62)

where 5/3 < p ≤ 2 and c = 2, …, m.

Therefore, for all 5/3 < p ≤ 2

γ_{n}(p) ≤ \frac{p}{p − 5/3} ∑_{c=2}^{m} \binom{m}{c}\{B_{c}(0, δ_{c})n^{− \frac{1}{4} (3p−5)} + B_{c}(\sqrt{n}, δ_{c})\}\frac{1}{\sqrt{n}}.  (63)

In (3) we can put p = 35/18. Note that for every x > 0

B_{c}(0, δ_{c}) ≤ x^{δ_{c}} + B_{c}(x, δ_{c}).  (64)

Assume x = n^{1/8}. Then from (63) and (64) it can be deduced

γ_{n}(35/18) ≤ 7 ∑_{c=2}^{m} \binom{m}{c}\{(n^{δ_{c}/8} + B_{c}(n^{1/8}, δ_{c}))n^{−5/32} + B_{c}(\sqrt{n}, δ_{c})\}\frac{1}{\sqrt{n}}
≤ 7 · 2^{m}E|g_{1}|^{3} \frac{1}{\sqrt{n}} + 14 ∑_{c=2}^{m} \binom{m}{c}B_{c}(n^{1/8}, δ_{c})\frac{1}{\sqrt{n}},  (65)

because 1 ≤ E|g_{1}|^{3} and B_{c}(\sqrt{n}, δ_{c}) ≤ B_{c}(n^{1/8}, δ_{c}). As a result, from (3), (53), (61) and (65) the inequality (5) can be obtained.

Proof of Theorem 4. To refine the estimate for ζ_{4}(t) given by (43). According to (36) the following representation of ζ_{4}(t) can be written

ζ_{4}(t) = \frac{(m − 1)\sqrt{n}}{2σ_{n}} · φ^{n−2}(\frac{t}{σ_{n}\sqrt{n}})E\{g_{n2}(X_{1}, X_{2}) ∏_{s=1}^{2}(e^{itξ_{s}} − 1)\} + R_{n6}(t), (66)

where

R_{n6}(t) = \frac{\sqrt{n}}{σ_{n}m} ∑_{c=3}^{m} \binom{m}{c}φ^{n−c} (\frac{t}{σ_{n}\sqrt{n}}) E\{g_{nc}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{itξ_{s}} − 1)\}.

First R_{n6}(t) has to be estimated. According to the estimate |e^{ix}−1| ≤ |x|, x ∈ R, it can be written

|E{g_{nc}(X_{1}, …, X_{c}) ∏_{s=1}^{c}(e^{itξ_{s}} − 1)}| ≤ |t|^{c}σ_{n}^{−c}n^{−c/2}E|g_{nc}| ∏_{s=1}^{c} |Z_{s}|,  (67)

where

Z_{s} = g_{n1}(X_{s}) + ∑_{c=2}^{m} \binom{m}{c}\frac{c}{m}h_{nc}(X_{s}).

According to Hölder’s inequality it can be written as

E|g_{nc}| ∏_{s=1}^{c} |Z_{s}| ≤ (E|g_{nc}|^{3/2})^{2/3} (E|Z_{1}|^{3})^{c/3} .

Reconciling (17) with (61)

E|Z_{1}|^{3} ≤ 8E|g_{1}^{n}|^{3} + 2^{2m+6}\sqrt{n}β_{n}
≤ 8E|g_{1}|^{3} + (2m + 1)2^{2m+5} ∑_{c=2}^{m} \binom{m}{c}B_{c}(n^{(2c−1)/2}, δ_{c}),

and, then, according to (26) can be obtained:

E|g_{nc}|^{3/2} ≤ ((c + 1)!)^{1/2} · 2^{c} · E|g_{c}|^{3/2}I(|g_{c}| ≤ n^{(2c−1)/2}).

If 3 ≤ c ≤ m, then δ_{c} < 3/2. Therefore (56) can be applied assuming p = 3/2. Hence

E|g_{nc}|^{3/2} ≤ ((c + 1)!)^{1/2} \frac{2^{c} · 3(2c − 1)}{2c − 5}B_{c}(0, δ_{c})n^{(2c−5)/4}.

Consequently,

|R_{n6}(t)| ≤ \{∑_{c=3}^{m} \binom{m}{c}c! 2^{c+1}(|t|^{c} · |φ(\frac{t}{σ_{n}\sqrt{n}})|)σ_{n}^{−c−1}B_{c}^{2/3}(0, δ_{c})

×(8E|g_{1}|^{3} + (2m + 1)2^{2m+5} ∑_{d=2}^{m} \binom{m}{d}B_{d}(n^{(2d−1)/2}, δ_{d}))^{c/3}n^{− \frac{c−1}{6}} \}\frac{1}{\sqrt{n}}. (68)

To make an estimate for the first term on the right-hand side of (66) we assume c = 2 in (37) and (38). Hence,

|E{g_{n2}(X_{1}, X_{2}) ∏_{s=1}^{2}(e^{itξ_{s}} − 1)}|

≤ |E\{g_{2}^{n}(X_{1}, X_{2}) ∏_{s=1}^{2}(e^{\frac{it}{σn\sqrt{n}} (g_{1}^{n}(X_{s})+h̄^{n}(X_{s}))} − 1)\}| + 4E|g_{2}|I(|g_{2}| > n^{\frac{3}{2}} ).

According to (39) when c = 2

|E\{g_{2}^{n}(X_{1}, X_{2}) ∏_{s=1}^{2}(e^{\frac{it}{σn\sqrt{n}} (g_{1}^{n}(X_{s})+h̄^{n}(X_{s}))} − 1)\}|

≤ t^{2}n^{−1}σ_{n}^{−2}E|g_{2}^{n}||g_{1}^{n}(X_{1})||g_{1}^{n}(X_{2})| + 2t^{2}n^{−1}σ_{n}^{−2}E|g_{2}^{n}||h̄^{n}(X_{1})||g_{1}^{n}(X_{2})|

+t^{2}n^{−1}σ_{n}^{−2}E|g_{2}^{n}||h̄^{n}(X_{1})||h̄^{n}(X_{2})|.

Applying (43), Hölder’s inequality and the equality Eg_{1}^{2} = 1 we can obtain

E|g_{2}^{n}||h̄^{n}(X_{1})||g_{1}^{n}(X_{2})| ≤ (E|g_{2}^{n}|^{2})^{1/2} 2^{m/2}β_{n}^{1/2},
E|g_{2}^{n}||h̄^{n}(X_{1})||h^{n}(X_{2})| ≤ (E|g_{2}^{n}|^{2})^{1/2} 2^{m}β_{n}.

Combining these inequalities with (66) for all t ∈ R it appears that

|ζ_{4}(t)| ≤ \frac{(m − 1)}{2\sqrt{n}}σ_{n}^{−3}t^{2} |φ (\frac{t}{σ_{n}\sqrt{n}})|^{n−2} E|g_{2}^{n}||g_{1}^{n}(X_{1})||g_{1}^{n}(X_{2})|

+\frac{(m − 1)2^{m/2}}{\sqrt{n}} σ_{n}^{−3}t^{2} |φ (\frac{t}{σ_{n}\sqrt{n}})|^{n−2} (E|g_{2}^{n}|^{2})^{1/2} β_{n}^{1/2}

+\frac{(m − 1)2^{m−1}}{\sqrt{n}} σ_{n}^{−3}t^{2} |φ (\frac{t}{σ_{n}\sqrt{n}})|^{n−2} (E|g_{2}^{n}|^{2})^{1/2} β_{n}

+\frac{2(m − 1)}{σ_{n}} |φ (\frac{t}{σ_{n}\sqrt{n}})|^{n−2} \{\sqrt{n}E|g_{2}|I(|g_{2}| > n^{3/2})\} + R_{n6}(t),  (69)

where R_{n6}(t) satisfies (68).

Referring to (59) when c = 2

E|g_{2}||g_{1}(X_{1})||g_{1}(X_{2})| ≤ (1 + 9B_{2}(0, 5/3))^{2/3} · (E|g_{1}|^{3})^{2/3}.  (70)

The inequality (6) is obtained as a result of the application of (69) to (45).

Proof of Theorem 5. We denote

Φ_{n}(x) = F (x) + \frac{κ_{3}}{6} · \frac{1}{\sqrt{2πn}}(1 − x^{2})e^{− \frac{x^{2}}{2}} , x ∈ R.

It can be verified that

ϕ_{n}(t) = ∫_{−∞}^{∞} e^{itx}dΦ_{n}(x) = e^{− \frac{t^{2}}{2}} (1 + \frac{κ_{3}}{6} (it)^{3} \frac{1}{\sqrt{n}}), t ∈ R.

Employing Esseen’s inequality

sup_{x} |F_{n}(x) − Φ_{n}(x)| ≤ ∫_{−\sqrt{n} ln n}^{\sqrt{n} ln n} |φ_{n}(t) − ϕ_{n}(t)||t|^{−1}dt + c(1 + |κ_{3}|)(\sqrt{n} ln n)^{−1}, (71)

where c > 0 is an absolute constant.

Estimate the integral in formula (71) applying (18) with p = 15/8 and ε = 1/16. First represent this integral as a sum

∫_{−\sqrt{n} ln n}^{\sqrt{n} ln n} |φ_{n}(t) − ϕ_{n}(t)||t|^{−1}dt = I_{n} + J_{n},  (72)

where

I_{n} = ∫_{−\sqrt{n}/a}^{\sqrt{n}/a} |φ_{n}(t) − ϕ_{n}(t)||t|^{−1}dt,
J_{n} = ∫_{\frac{\sqrt{n}}{a} ≤|t|≤\sqrt{n} ln n} |φ_{n}(t) − ϕ_{n}(t)||t|^{−1}dt.

Estimate of J_{n}. According to the definition a = E|Z_{1}|^{3}σ_{n}^{−3/2}. Further, consider

E|g_{n1}|^{3} − E|Z_{1}|^{3} = E(|g_{n1}| − |Z_{1}|)(|g_{n1}|^{2} + |g_{n1}||Z_{1}| + |Z_{1}|^{2}).

Taking into account the inequalities

|g_{n1}| ≤ 2\sqrt{n}, |Z_{1}| ≤ 2\sqrt{n} ∑_{c=1}^{m} \binom{m}{c}\frac{c}{m},

and (54) we get

|E|g_{n1}|^{3} − E|Z_{1}|^{3}| ≤ 4n ∑_{c=2}^{m} \binom{m}{c}\frac{c}{m}E|h_{nc}(X_{1})|(1 + ∑_{c=1}^{m} \binom{m}{c}\frac{c}{m})^{2}
≤ 2^{2m+3} ∑_{c=2}^{m} \binom{m}{c}\{nE|g_{c}|I(|g_{c}| > n^{(2c−1)/2})\}
≤ 2^{2m+2} ∑_{c=2}^{m} \binom{m}{c}(2c + 1)B_{c}(n^{(2c−1)/2}, δ_{c}).  (73)

Using analogous reasoning

|E|g_{n1}|^{3} − E|g_{1}^{n}|^{3}| ≤ 7E|g_{1}|^{3}I(|g_{1}| > \sqrt{n})  (74)

¿From (73), (74) and the formula

E|g_{1}^{n}|^{3} = E|g_{1}|^{3} − E|g_{1}|^{3}I(|g_{1}| > \sqrt{n})

it follows that

−b_{n} + E|g_{1}|^{3} ≤ E|Z_{1}|^{3} ≤ E|g_{1}|^{3} + b_{n},  (75)

where

b_{n} = 2^{2m+2} ∑_{c=1}^{m} \binom{m}{c}(2c + 1)B_{c}(n^{(2c−1)/2}, δ_{c}).

Taking into account that b_{n} → 0 as n → ∞ the following inequality

b_{n} ≤ \frac{1}{2}E|g_{1}|^{3},  (76)

holds for sufficiently large n. Therefore, from (75) it can be obtained

\frac{1}{2}E|g_{1}|^{3} ≤ E|Z_{1}|^{3} ≤ E|g_{1}|^{3}  (77)

for n → ∞. Analogously to the deduction of (47) and using (76)

σ_{n}^{2} ≥ 1 − \frac{2}{\sqrt{n}}b_{n} ≥ 1 − \frac{1}{\sqrt{n}}E|g_{1}|^{3}.

Consequently,

σ_{n}^{2} ≥ \frac{1}{2}  (78)

for sufficiently large n: \sqrt{n} ≥ 2E|g_{1}|^{3}.

Further, consider φ(\frac{t}{σ_{n}\sqrt{n}}) in the domain \frac{\sqrt{n}}{a} ≤ |t| ≤ \sqrt{n} ln n. The following representation can be written

φ(\frac{t}{σ_{n}\sqrt{n}}) = E exp (\frac{itg_{1}(X_{1})}{σ_{n}\sqrt{n}})

+E exp (\frac{itg_{n1}(X_{1})}{σ_{n}\sqrt{n}}) \{exp (\frac{it}{σ_{n}\sqrt{n}} ∑_{c=2}^{m} \binom{m}{c}\frac{c}{m}h_{nc}(X_{1})) − exp (\frac{itḡ_{n1}(X_{1})}{σ_{n}\sqrt{n}})\}

where ḡ_{n1}(X_{1}) = g_{1}(X_{1})I(|g_{1}(X_{1})| > \sqrt{n}) − Eg_{1}(X_{1})I(|g_{1}(X_{1})| > \sqrt{n}). With the help of the inequality |e^{ix} − 1| ≤ |x|, x ∈ R, it can be obtained

|φ(\frac{t}{σ_{n}\sqrt{n}})| ≤ |χ(\frac{t}{σ_{n}\sqrt{n}})| + \frac{2|t|}{σ_{n}\sqrt{n}}E|g_{1}|I(|g_{1}| > \sqrt{n})

+\frac{2|t|}{σ_{n}\sqrt{n}} ∑_{c=2}^{m} \binom{m}{c}\frac{c}{m}E|g_{c}|I(|g_{c}| > n^{(2c−1)/2})

≤ |χ(\frac{t}{σ_{n}\sqrt{n}})| + \frac{2|t|}{σ_{n}\sqrt{n}} \frac{b_{n}}{n} ,  (79)

where χ(t) = E exp(itg_{1}(X_{1})), t ∈ R. ¿From (76)–(79) it can be concluded that

sup_{\frac{\sqrt{n}}{a} ≤|t|≤\sqrt{n} ln n} |φ(\frac{t}{σ_{n}\sqrt{n}})| ≤ sup_{|t|≥_{\frac{1}{2E|g_{1}|^{3}}}} |χ(t)| + \frac{ln n}{\sqrt{n}}.  (80)

Further due to p = 15/8 and ε = 1/16 in (18) k ≥ n^{1/34} for n → ∞. According to the Cramer condition of this theorem sup_{|t|≥_{\frac{1}{2E|g_{1}|^{3}}}} |χ(t)| < 1. Therefore for every s ≥ 1

( sup_{\frac{\sqrt{n}}{a} ≤|t|≤\sqrt{n} ln n} |φ(\frac{t}{σ_{n}\sqrt{n}})|)^{k} = O(n^{−s})  (81)

according to (80) as n → ∞. Furthermore, it is clear that

J_{n} ≤ ∫_{\frac{\sqrt{n}}{2E|g_{1}|^{3}} ≤|t|≤\sqrt{n} ln n} |φ_{n}(t) − ϕ_{n}(t)||t|^{−1}dt
≤ ∫_{\frac{\sqrt{n}}{2E|g_{1}|^{3}} ≤|t|≤\sqrt{n} ln n} |φ_{n}(t)||t|^{−1}dt + ∫_{\frac{\sqrt{n}}{2E|g_{1}|^{3}} ≤|t|≤\sqrt{n} ln n} |ϕ_{n}(t)||t|^{−1}dt.

Due to (44) and (81)

∫_{\frac{\sqrt{n}}{2E|g_{1}|^{3}} ≤|t|≤\sqrt{n} ln n} |φ_{n}(t)||t|^{−1}dt = o(n^{−1/2}).

According to the property of function ϕ_{n}(t)

∫_{\frac{\sqrt{n}}{2E|g_{1}|^{3}} ≤|t|≤\sqrt{n} ln n} |ϕ_{n}(t)||t|^{−1}dt = o(n^{−1/2}).

Consequently,

J_{n} = o(n^{−1/2}).  (82)

Estimate of I_{n}. The function ϕ_{n}(t) can be represented as a sum ϕ_{n}(t) = ϕ_{n1}(t) + ϕ_{n2}(t), where

ϕ_{n1}(t) = e^{−t^{2}/2}(1 + (it)^{3} \frac{1}{6} · E g_{1}^{3} · \frac{1}{\sqrt{n}}),

ϕ_{n2}(t) = e^{−t^{2}/2}(it)^{3} \frac{1}{2}(m − 1)E(g_{2}(X_{1}, X_{2})g_{1}(X_{1})g_{1}(X_{2})) · \frac{1}{\sqrt{n}}.

Therefore, according to (44)

φ_{n}(t) − ϕ_{n}(t) = (φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − ϕ_{n1}(t)) + (itζ_{4}(t) − ϕ_{n2}(t)) + R_{n7}(t)

in the domain |t| ≤ \sqrt{n}/a where

R_{n7}(t) = R_{n1}(t) + R_{n2}(t) + it(ζ_{1}(t) + ζ_{2}(t)) + itφ^{k}(\frac{t}{σ_{n}\sqrt{n}})(R_{n3}(t) + R_{n4}(t)).

Therefore the following inequality is true

I_{n} ≤ I_{n1} + I_{n2} + I_{n3} + I_{n4}  (83)

where

I_{n1} = ∫− \frac{\sqrt{n}}{2\sqrt{b}}^{\frac{\sqrt{n}}{2\sqrt{b}}} |φ^{n}(\frac{t}{σ_{n}\sqrt{n}})−ϕ_{n1}(t)||t|^{−1}dt, I_{n2} = ∫_{\frac{\sqrt{n}}{2\sqrt{b}} ≤|t|≤ \frac{\sqrt{n}}{a}} |φ^{n}(\frac{t}{σ_{n}\sqrt{n}})−ϕ_{n1}(t)||t|^{−1}dt,

I_{n3} = ∫− \frac{\sqrt{n}}{a}^{\frac{\sqrt{n}}{a}} |itζ_{4} − ϕ_{n2}(t)||t|^{−1}dt, I_{n4} = ∫− \frac{\sqrt{n}}{a}^{\frac{\sqrt{n}}{a}} |R_{n7}(t)||t|^{−1}dt.

Here

b = σ_{n}^{−2}E|Z_{1}|^{4}.

Note that a ≤ \sqrt{b} is always true. Consider every summand on the right-hand side of (83). First have a look at I_{n1}. ¿From Bhattacharya and Ranga Rao (1986) it is known that the following inequality is true

|φ^{n}(\frac{t}{σ_{n}\sqrt{n}}) − exp(−\frac{1}{2}t^{2}) (1 + \frac{(it)^{3}}{σ\sqrt{n}}EZ_{1}^{3})| ≤ \frac{b}{n}t^{4}(1 + t^{2}) exp (−\frac{t^{2}}{4} ) .  (84)

Here

E|Z_{1}|^{4} ≤ 8 (E|g_{n1}|^{4} + E| ∑_{c=2}^{m} \binom{m}{c}\frac{c}{m}h_{n}(X_{1})|^{4})

≤ 8(8E|g_{1}|^{4}I(|g_{1}| ≤ \sqrt{n})) + (m − 1)^{3} ∑_{c=2}^{m} \binom{m}{c}^{4}Eh_{nc}^{4}(X_{1})

≤ 64E|g_{1}|^{4}I(|g_{1}| ≤ \sqrt{n}) + 64(m − 1)^{3} ∑_{c=2}^{2} \binom{m}{n}^{4}\{n^{3/2}E|g_{c}|I(|g_{c}| > n^{(2c−1)/2})\},

that is

\frac{1}{n}E|Z_{1}|^{4} ≤ 64n^{−1}E|g_{1}|^{4}I(|g_{1}| ≤ \sqrt{n}) + 64m^{4m}β_{n}.

Substituting (84) into I_{n1} and taking into account (75) and (78), it can be observed that I_{n1} = o(n^{−1/2}) as n → ∞. To estimate I_{n2} it should be noted that

\frac{\sqrt{n}}{2\sqrt{b}} ≥ \frac{n^{1/4}}{2\sqrt{3} 2^{m}E|g_{1}|^{3}} .

According to (15) I_{n2} = o(n^{−1/2}), for n → ∞. To estimate the I_{n3}, formula (66) is applied. Using (66) it can be written: I_{n3} ≤ I_{n3}^{1} + I_{n3}^{2}, where

I_{n3}^{1} = ∫− \frac{\sqrt{n}}{a}^{\frac{\sqrt{n}}{a}} |itf_{n}(t) − ϕ_{n2}(t)||t|^{−1}dt, I_{n3}^{2} = ∫− \frac{\sqrt{n}}{a}^{\frac{\sqrt{n}}{a}} |R_{n6}(t)|dt,

and

f_{n}(t) = \frac{(m − 1)\sqrt{n}}{2σ_{n}}φ^{n−2}(\frac{t}{σ_{n}\sqrt{n}})E\{g_{n2}(X_{1}, X_{2}) ∏_{s=1}^{2}(e^{itξ_{s}} − 1)\}.

Consequently, using (68) and taking (14) I_{n3}^{2} = o(n^{−1/2}).

According to (15) integral I_{n3}^{1} can be estimated in the following way

I_{n3}^{1} ≤ \frac{(m − 1)}{2} ∫− \frac{\sqrt{n}}{a}^{\frac{\sqrt{n}}{a}} |δ_{n}(t)| exp (−\frac{t^{2}}{2} ) dt + o(n^{−1/2}),

where

δ_{n}(t) = \frac{\sqrt{n}}{σ_{n}}E \{g_{n2}(X_{1}, X_{2}) ∏_{s=1}^{2} (e^{\frac{itZs}{σn\sqrt{n}}} − 1)\} − E (g_{2}(X_{1}, X_{2})g_{1}(X_{1})g_{1}(X_{2})) \frac{(it)^{2}}{\sqrt{n}} .

To analyze δ_{n}(t), the formula

e^{ix} − 1 = ix(1 + ∫_{0}^{1}(e^{ixu} − 1)du)

for x = tZ_{s}/(σ_{n}\sqrt{n}), s = 1, 2, is applied. Therefore,

E \{g_{n2}(X_{1}, X_{2}) ∏_{s=1}^{2} (e^{\frac{itZs}{σn\sqrt{n}}} − 1)\} = \frac{(it)^{2}}{σ_{n}^{2}n} E(g_{n2}(X_{1}, X_{2})Z_{1} · Z_{2})

+\frac{2(it)^{2}}{σ_{n}^{2}n} ∫_{0}^{1} duE (g_{n2}(X_{1}, X_{2})Z_{1} · Z_{2} (e^{\frac{itZ_{1}}{σn\sqrt{n}} u} − 1))

+\frac{(it)^{2}}{σ_{n}^{2}n} ∫_{0}^{1} du_{1} ∫_{0}^{1} du_{2}E (g_{n2}(X_{1}, X_{2})Z_{1} · Z_{2} (e^{\frac{itZ_{1}}{σn\sqrt{n}} u_{1}} − 1) (e^{\frac{itZ_{2}}{σn\sqrt{n}} u_{2}} − 1)) .

Substituting this formula into δ_{n}(t) and applying the inequality |e^{ix} − 1| ≤ 2|x|^{q}, 0 < q < 1, we obtain

|δ_{n}(t)| ≤ \frac{t^{2}}{\sqrt{n}} \{|\frac{1}{σ_{n}^{3}} E (g_{n2}(X_{1}, X_{2})Z_{1} · Z_{2}) − E (g_{2}(X_{1}, X_{2})g_{1}(X_{1}) · g_{1}(X_{2}))|

+4|t|^{q}σ_{n}^{−2−q}n^{−q/2}E (|g_{n2}(X_{1}, X_{2})||Z_{1}|^{1+q} · |Z_{2}|)
+4|t|^{2q}σ_{n}^{−2−2q}n^{−q}E (|g_{n2}(X_{1}, X_{2})||Z_{1} · Z_{2}|^{1+q})} .  (85)

Assuming q = 1/2 in (85) and taking into account (78) the following relations can be obtained

|\frac{1}{σ_{n}^{3}} E (g_{n2}(X_{1}, X_{2})Z_{1} · Z_{2}) − E (g_{2}(X_{1}, X_{2})g_{1}(X_{1}) · g_{1}(X_{2}))| → 0,  (86)
n^{−1/4}E (|g_{n2}(X_{1}, X_{2})||Z_{1}|^{3/2} · |Z_{2}|) → 0,  (87)
n^{−1/2}E (|g_{n2}(X_{1}, X_{2})||Z_{1} · Z_{2}|^{3/2}) → 0.  (88)

Indeed, relation (86) follows from the representations

E (g_{n2}(X_{1}, X_{2})Z_{1} · Z_{2}) = Eg_{2}^{n}(X_{1}, X_{2})g_{n1}(X_{1})g_{n1}(X_{2})
+2g_{2}^{n}(X_{1}, X_{2})g_{n1}(X_{1})h̄^{n}(X_{1}) + Eg_{2}^{n}(X_{1}, X_{2})h̄^{n}(X_{1})h̄^{n}(X_{2}),

\frac{1}{σ_{n}^{3}} = 1 + (1 − σ_{n}^{2})\frac{1 + σ_{n} + σ_{n}^{2}}{σ_{n}^{3}(1 + σ_{n})} ,

the estimates (47), (48) and the Hölder inequality.

Relations (87) and (88) can be deduced from the following estimates which are obtained with the help of the Hölder inequality

n^{−1/4}E|g_{n2}(X_{1}, X_{2})||Z_{1}|^{3/2}|Z_{2}| ≤ n^{−1/4} (E|g_{n2}(X_{1}, X_{2})|^{2})^{1/2} (E|Z_{1}|^{3} · E|Z_{2}|^{2})^{1/2}
≤ (\frac{1}{\sqrt{n}}E|g_{2}^{n}|^{2})^{1/2} (E|Z_{1}|^{3})^{5/6} → 0,

n^{−1/2}E|g_{n2}(X_{1}, X_{2})||Z_{1}|^{3/2}|Z_{2}|^{3/2} ≤ n^{−1/2} (E|g_{n2}(X_{1}, X_{2})|^{2})^{1/2} E|Z_{1}|^{3}
≤ n^{−1/4} (\frac{1}{\sqrt{n}}E|g_{2}^{n}|^{2})^{1/2} E|Z_{1}|^{3} → 0.

Consequently, I_{n3}^{1} = o(n^{−1/2}) for n → ∞. Estimate I_{n3}^{2} = o(n^{−1/2}) for n → ∞ follows from (14) and (68). Therefore I_{n3} = o(n^{−1/2}). In (45) it was proved that I_{n4} = o(n^{−1/2}). Hence,

I_{n} = o(n^{−1/2}) as n → ∞.  (89)

From (71), (72), (82) and (89) the relation (7) can be obtained. Therefore, Theorem 5 is proved.

Acknowledgement.

The author would like to thank the staff of the School of Mathematics and Statistics at the University of Sydney for their kind hospitality during his stay. The author’s visit to the University of Sydney was supported by an Australian Research Council Grant.

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