Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

: Asymptotic expansions


Introduction
Asymptotic expansions for symmetric statistics are studied by many people. See, e.g., Callaert-Janssen-Veraverbeke (1980) [1], Withers (1988) [2], Maesono (2004) [3], and so on. They treat U-statistics with non-degenerate kernels. On the other hand, Bentkus-Götze (1999) [4]and Zubayraev (2011) [5] obtained optimal bounds in asymptotic expansions for U-statistics with degenerate kernels. They treat the following modified U-statistics, where φ(·, ·) is a symmetric function, φ 1 (·) is a measurable function and {ξ i } are i.i.d. random variables. W n coincides with V-statistics when If φ 1 (x) = 0 for any x, then W n coincides with U-statistics. They obtained asymptotic expansions with remainder O(n −1 ) for the distribution function of W n .In this paper, we investigate asymptotic expansions for the simple U-statistics and the V-statistics with degree two defined by respectively. We obtain asymptotic expansions with remainder O(n 1−p/2 ) for some p ≥ 4 for the distribution function of U n or V n under some assumptions for {ξ i } and u(x, y). Our scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of u ξ i , ξ j . The key condition for the convergence is the nuclearity of a linear operator T u defined by the kernel function u(x, y). The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables due to Sazonov-Uyanov (1995) [6].

Symmetric Statistics
Let ξ j , j ≥ 1 be i.i.d. random variables with a probability distribution µ on an arbitrary measurable space (X, B). Suppose that u(x 1 , x 2 , · · · , x n ) is a real valued symmetric function for some k ≥ 1, i.e., for any permutation (i 1 , i 2 , · · · , i k ) of (1, 2, · · · , k). A statistics defined by the kernel function u(x 1 , x 2 , · · · , x k ) is called a symmetric statistics. The followings are the typical examples of the symmetric statistics.
Example 1. U-statistics with degree k ≥ 1: Example 2. V-statistics with degree k ≥ 1: In this paper, we treat V-statistics V n and U-statistics U n with degree two defined by (3) when the kernel function u(x, y) is degenerate, i.e., for any real number x.

Non-Central Limit Theorems for U-Statistics with Degenerate Kernels
Assume that {ξ i } are i.i.d. random variables with a distribution µ. Let u(x, y) be a real valued symmetric function on R × R and square integrable such that Suppose that u(x, y) is a degenerate kernel satisfying the condition (7). Let L 2 (R, µ) be the space of all square integrable functions with respect to µ. Then, according to Serfling (1980) [7], we see that the kernel u(x, y) induces a bounded linear operator L 2 (R, µ) → L 2 (R, µ) (trace class) defined by which has eigenvalues {λ i } and eigenfunctions {g i } satisfying for each i ≥ 1 With respect to (10), see Serfling (1980) for each i, j ≥ 1. Serfling (1980) [7] showed the non-central limit theorem for U-statistics with degree 2.
It is well known that the rate of convergence in (13) is Serfling (1980) [7] for more details). We obtain asymptotic expansions for U n and V n using asymptotic expansions due to Sazonov-Uyanov (1995) [6] for sums of Hilbert space valued i.i.d. random variables in the next section.

Asymptotic Expansions for Single Sums which Hit a Ball in a Hilbert Space
In this section we consider an asymptotic expansions for sums of Hilbert space valued random vectors {X i } according to Sazonov-Uyanov (1995) [6]. Let {X i } be a sequence of i.i.d. random vectors in a separable Hilbert space H with E[X 1 ] = 0 and E X 1 2 = 1, where x 2 = x, x for x ∈ H and ·, · is the inner product in H. Define the covariance operator V of X 1 by for x, y ∈ H. Denote by σ 2 1 ≥ σ 2 2 ≥ · · · the eigenvalues of V and by e 1 , e 2 , · · · be the orthonormal eigenvectors corresponding to the eigenvalues. Put where Put for any L > 0. Let Y be the H-valued Gaussian random variables with mean 0 and the covariance operator V. For a, h ∈ H, r > 0, i = 0, 1, · · · we put Define the differential operators d k h by Put and χ j = χ j,0 . For positive integers l 1 , l 2 , · · · , l s we put and for integers k where l (j) = l 1 ! · · · l j ! and ∑ denotes the summation over all, such that The following theorem is the key result for the proofs of our theorems.
In addition, the terms in the asymptotic expansion for ε > 0 satisfies the estimates for even i, and if i is odd, then we have

The Sato-Mercer Theorem
In the proofs of our theorems we use the Fourier series expansion for the kernel function u ξ i , ξ j by eigenvalues and eigenfunction of the linear operator T u defined by (9). Since (11) holds in the sense of the L 2 -convergence, (11) can not be applied to show the asymptotic expansions for U-statistics or V-statistics as it is. We show that u ξ i , ξ j can be represented by the Fourier series expansion in (11) almost surely using the following Sato-Mercer theorem. (See Sato (1992) [8] for more details.)

Theorem 3. (The Sato-Mercer theorem)
Let X be a separable metric space with a Borel measure ν on X, and K(x, y) be a function on X × X such that there exists a Borel-measurable subset X 0 , such that ν(X\X 0 ) = 0. (34) Suppose that K(x, y) is continuous on X 0 and satisfies X X for any f ∈ L 2 (X, ν). Then, the linear operator T K on L 2 (X, ν) defined by is nuclear if, and only if, holds.
From Theorem 3, we have the next result.

Theorem 4.
Let ξ j , j ≥ 1 be i.i.d. random variables with the distribution µ. Let u(x, y) be a real valued symmetric function on R × R and T u be a linear operator defined by Suppose that u(x, y) is the square integrable degenerate kernel of the linear operator T u , such that for any f ∈ L 2 (R, µ) and for any x ∈ R. Let {λ k } and {g k } be eigenvalues and eingenfunctions of the linear operator T u , respectively. Suppose Furthermore assume that there exists a Lebesgue measurable subset X 0 ⊂ R, such that and u(x, y) is continuous on X 0 . Then, we have for each i, j ≥ 1.

Remark 1.
If the symmetric function u(x, y) is piecewise continuous on R, then there exists X 0 ⊂ R satisfying (44) such that u(x, y) is continuous on X 0 . In the next section, we show a typical example of U-or V-statistics defined by such piecewise continuous function u(x, y) as its kernel function.

Asymptotic Expansions for Degenerate V-Statistics and U-Statistics with Degree 2
For applying Theorem 2 for Hilbert space valued random variables to the proof of asymptotic expansions for V n , we represent V n by sums of Hilbert space valued random variables {G i } by the following method.
According to K.-Yoshihara (1994) [9], we introduce a separable Hilbert space Hequipped with the inner product ·, · and the norm · as follows, x, y = ∞ ∑ k=1 |λ k |x k y k (52) and Using the assumptions of Theorem 4, we have from (10) and (48) that which implies that we can define H-valued random variables by for each i ≥ 1. Let {U n , n ≥ 1} and {V n , n ≥ 1} be U-statistics and V-statistics with degree 2 defined by (3), respectively.

Theorem 5.
Without loss of generality we assume that θ = 0. Suppose that ξ j , j ≥ 1 is a sequence of i.i.d. random variables with the distribution µ. Assume that u(x, y) is a square integrable symmetric function with respect to µ × µ satisfying (40) ∼ (42). Suppose that for some Furthermore, without loss of generality, assume that Let Y be the H-valued Gaussian random variables with mean 0 and the covariance operator V satisfying (14) with the eigenvalues σ 2 1 ≥ σ 2 2 ≥ · · · and the orthogonal eigenvectors e 1 , e 2 , · · · . For any t ≥ 0, integer k ≥ 2, let L be a positive number, such that Then, for L ≤ n 1/2 and α ≥ 1 5 , and B j (0, r) = n −(j−2) / 2 G 1 j . (62)

Proof.
Put Recall that for each i, Then we have Thus, we can apply Theorem 2 to show Theorem 5.
Therefore, from (67) and (68) Since the right hand side of (69) is the V-statistics with the degenerate kernel v(x, y) satisfying all assumptions of Theorem 5, Theorem 6 holds from Theorem 5.

Cramer-Von Mises Statistics
There are some examples of U-statistics or V-statistics for which the above theorems are applicable under the assumption of nuclearity of the kernel functions where the above theorems are applicable. is nuclear from Theorem 3. Therefore, since the degenerate kernel u(x, y) defined by (70) satisfies all assumptions of Theorem 5, Theorem 5 holds for the Cramér-von Mises Statistics. Furthermore, Theorem 6 also holds for U-statistics with the degenerate kernel v(x, y) defined by (67) and (70).

Conclusions
Bentkus-Götze (1999) [4] and Zubayraev (2011) [5] obtained the remainder O(n −1 ) in asymptotic expansions for U-statistics or V-statistics with degenerate kernels. From Theorems 5 and 6, if we assume E G 1 p ≤ ∞, p ≥ 4 and some conditions, then we obtain the remainder O n 1−p / 2 . Applying Theorem 5, we obtain asymptotic expansions for the Cramér-von Mises statistics of the uniform distribution U(0, 1) with the remainder O n 1−p/2 for any p ≥ 4.
Funding: Grant-in-Aid Scientific Research (C), No.18K03431, Ministry of Education, Science and Culture, Japan.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The author uses no data.