A Constant on a Uniform Bound of a Combinatorial Central Limit Theorem

Let n be a positive integer and Y(i, j), i, j = 1, ..., n, be random variables with finite fourth moments. Let ? be a random permutation on {1, ..., n} which independent of Y(i, j)’s. In this paper, we use Stein’s method and the technique from (Laipaporn, K., 2008) to give a uniform bound in a combinatorial central limit theorem of W = n *i=1 Y(i, ?(i)). For a sufficient large n, we yield the rate 27.72 ?n . This constant is better than the result in (Neammanee, K., 2005). Keywords: Uniform bound, Combinatorial central limit theorem, Stein’s method, Random permutation

We define ( This paper is concerned with the normal approximation to the distribution of W which is always called a combinatorial central limit theorem.The literature concerning a combinatorial central limit theorem dates back to 1944 when Wald and Wolfowitz (Wald, A., 1944) first established the asymptotic normality of the statistic η n = n i=1 a i b π(i) where a i , b i , i = 1, 2, ..., n, are two sequences of real numbers and π is a random permutation of {1, 2, ..., n}.This was extended by Hoeffding (Hoeffding, W., 1951) where Y(i, j) are n 2 real numbers.Matoo (Matoo, M., 1957) showed that a Lindeberg-type condition is sufficient for the asymptotic normality of W n .The same condition was also shown to be necessary in the case of η n by Hajek (Hajek, J., 1961).In 1972, Robinson (Robinson, J., 1972) obtained necessary and sufficient conditions for the moments of η n to converge to those of a normal distribution and Kolchin and Chistyakov(Kolchin, V.F., 1973) considered a different η n where π is no longer uniform but attributes equal probabilities to only those permutations with one cycle.
It seem so far that only limit theorem when Y(i, j)'s are real numbers has been proved.In the case when Y(i, j)'s are any random variables, the estimations have been obtained by Von Bahr(Von Bahr, B., 1976) and Ho and Chen (Ho, ¢ www.ccsenet.org/jmr ISSN: 1916-9795 S.T., 1978), and a Berry-Esseen-type bound was obtained by Bolthausen (Bolthausen, E., 1984) for univariate linear statistics and Bolthausen and Gotze (Bolthausen, E., 1993) for multivariate statistics.In 2005, Neammanee and Suntornchost (Neammanee, K., 2005) gave the uniform rate by using Stein's method.In this work, we improve the constant by using the technique from (Laipaporn, K., 2008).Our constant is sharper than the result in (Neammanee, K., 2005)(C ≥ 198).
Our results are as follows: and where μ(i, j) = EY(i, j).Then for n ≥ 30, where Φ is the standard normal distribution, and 2. Proof of Main Result (Theorem 1.1)

Auxillary results
In this section, we give auxiliary results for proving our main theorem (Theorem 1.1).First, we need to bound EW 4 .
Lemma 2.1 If W is defined as in (1), then for n ≥ 30, where By the fact that n ≥ 30 and (3) we have ¢ www.ccsenet.org

Journal of Mathematics Research
September, 2009 and We observe that Thus Next, we bound A 4 .Note that where i 3 = i 1 or i 2 and j 3 = j 1 or j 2 .Recall that, for fixed i 1 , j 1 , and (6), we obtain ¢ www.ccsenet.org/jmrISSN: 1916-9795 in which we used the fact that n ≥ 30 in the last inequality.
In 1972, Stein (Stein, C.M., 1972) introduced a powerful and general method to obtain explicit bound for the error in the normal approximation to the distribution of a sum of dependent random variables.His technique was relied on elementary differantial equations.For each z ∈ R, the Stein's equation for normal distribution function is where w ∈ R. It is well-known that the solution g w of ( 10) is of the form (Stein, C.M., 1972, p.22-23). ( To apply Stein's method, we construct the coupling W of W as follows. Let I and K be uniformly distributed random variables on {1, ..., n}, (I, K) uniformly distributed on {(i, k)|i, k = 1, ..., n, i k} and assume that they are independent of π and Y(i, j)'s.Define where Note that ( W, W) is an exchangeable pair (Neammanee, K., 2005, p.261).Clearly, S 1 and S 2 have the same distribution and so do S 3 and S 4 .We observe that Thus S 1 , S 2 , S 3 , S 4 are identically distributed.
Lemma 2.2 Let g : R → R be a continuous and piecewise continuously differentiable function.Then and where and I is the indicator function.
Proof: Let A be the σ-algebra generated by We can used the idea of (Neammanee, K., 2005) to show that Let μ(i, π(k)) = μ(i, j) for π(k) = j.We observe that for i k ¢ www.ccsenet.org/jmr We note from (3) that So, we can conclude that which implies where We note from (2) that ¢ www.ccsenet.org

Journal of Mathematics Research
September, 2009 Thus Hence Lemma 2.3 Let w ∈ R and g w be the unique bounded solution of (10).With the notation and assumptions of Theorem 1.1 and δ 4 ≤ 0.047, we have that for n ≥ 30, EY 2 (i, j).
Proof: Let A be defined as in Lemma 2.2.By the same argument of Lemma 5 in (Loh, W.Y., 1996), we have To prove the lemma, it suffices to find appropriate bounds for the terms on the right-hand side of (19).For the sake of clarity, we shall break the proof down into two steps.
¢ www.ccsenet.org/jmrISSN: 1916-9795 Step 1.We will bound the first term on the right-hand side of ( 19).Hence, by ( 17) By the symmetry of S 1 and S 2 , By the same technique of (20) we can show that and from the symmetry between S 3 and S 4 , Hence we can conclude from ( 20), ( 21), ( 22) and ( 23) that Step 2. We will bound the second term on the right-hand side of ( 19).
Note that where we have used (17) in the last inequality.By the fact that and  This proves the lemma.