Edgeworth Expansions for a Sample Sum from a Finite Set of Independent Random Variables *

Let {X 1 , · · · , X N } be a set of N independent random variables, and let S n be a sum of n random variables chosen without replacement from the set {X 1 , · · · , X N } with equal probabilities. In this paper we give a one-term Edgeworth expansion of the remainder term for the normal approximation of S n under mild conditions.


Introduction and main results
Let {X 1 , • • • , X N } be a set of independent random variables, µ k = EX k , 1 ≤ k ≤ N .Let R = (R 1 , • • • , R N ) be a random vector independent of X 1 , • • • , X N , such that P (R = r) = 1/N ! for any permutation r = (r 1 , • • • , r N ) of the numbers 1, • • • , N , and put S n = n j=1 X R j , 1 ≤ n ≤ N , that is for a sum of n random variables chosen without replacement from the set {X 1 , • • • , X N } with equal probabilities.
In contrast to rich investigations for the case X k = µ k , 1 ≤ k ≤ N , are (nonrandom) real numbers, there are only a few results concerned with the asymptotics of general S n discussed in this paper.von Bahr (1972) showed that the distribution of S n / √ V arS n may be approximated by a normal distribution under certain mild conditions.The rate of the normal approximation has currently been established by Zhao, Wu and Wang (2004), in which the paper improved essentially earlier work by von Bahr (1972).Along the lines of Zhao, Wu and Wang (2004), this paper discusses Edgeworth expansions for the distribution of S n / √ V arS n .Throughout the paper, let for j = 1, 2, 3, 4, and Theorem 1. Suppose that α 1 = 0 and β 2 = 1.Then, for all 1 ≤ n < N , where C is an absolute constant, with Φ(x) being a standard normal distribution, and Property (1) improves essentially a result of Mirakhmedov (1983).The related result in Theorem 1 of Mirakhmedov (1983) depends on max 1≤k≤N EX 4 k .Note that it is frequently the case that This condition is quite restrictive since it takes away the most interesting case that the X k are all degenerate.
In this case, the property (1) reduces to where , which gives one of main results in Bloznelis (2000a, b).
We next give a result complementary to Theorem 1.The result is better than Theorem 1 under certain conditions such as some of the X k 's are non-degenerate random variables and q is close to 0. Theorem 2. Suppose that α 1 = 0 and β 2 = 1.Then, for all 1 ≤ n ≤ N , where C is an absolute constant, G n (x), L 0 and δ 0 are defined as in Theorem 1, ∆ 2n = (nb 2 ) −1 β 4 and Ee itX k .
In the next section, we prove the main results.Throughout the paper we shall use C, C 1 , C 2 , ... to denote absolute constants whose value may differ at each occurrence.Also, I(A) denotes the indicator function of a set A, ♯(A) denotes the number of elements in the set A, k denotes N k=1 , and k denotes N k=1 .The symbol i will be used exclusively for √ −1.

Proofs of Theorems
Let µ k = EX k and Ψ(t) = E exp{itS n / √ nb}.Recall α 1 = 0 and β 2 = 1.As in (4) of Zhao, Wu and Wang (2004), where The main idea of the proofs is outlined as follows.We first provide the expansions and the basic properties for k Eρ k (ψ, t) in Lemmas 1-4.In Lemma 5, the idea in von Bahr (1972) is extended to give an expansion of Ψ(t) for the case n/N ≥ 1/2.The proofs of Theorems 1 and 2 are finally completed by virtue of the classical Esseen's smoothing lemma.
Then, for |u| ≤ where As in von Bahr (1972), we have where In view of (28) of Zhao, Wu and Wang (2004), for r > 0, C N,n,r ≤ 1, and for n ≥ 4 and r ≤ n, To prove (25) by using (26), we need some preliminary results. Write We have that β 4 ≥ 1 and by Taylor's expansion, for |u| ≤ 1  16 where |R 1k (u)| ≤ (1 + β 4k )u 4 /n 2 .Furthermore, by noting that since n/N ≥ 1/2 and |u| ≤ 1 16 (n/β 4 ) 1/4 , we have where where where A 2 is defined as in (4).As in the proof of ( 6), we may obtain This together with (38) yields, for |u| ≤ We are now ready to prove (25) by using (26).Rewrite (26) as where where the summation in the expression of I 1 is over all i j ≥ 0, j = 1, 2, 3 and i j > 0 for at least one j = 4, • • • , n.As in Mirakhmedov (1983), it follows from (36)-(37) that As for I 2 , it follows easily from ( 27) and (37) that We next estimate I 3 .Recalling that b = 1 − pα 2 and noting that I 3 = e P 3 j=1 p j B j , we have where L(u) = 3 j=1 p j B j − pα 2 u 2 /2 and we have used (38)-(39).Combining (40) and all above facts for I 1 -I 3 , we obtain which implies (25).The proof of Lemma 5 is now completed.2 After these preliminaries, we are now ready to prove the theorems.
Proof of Theorem 1.Without loss of generality, assume that nq > 256 and ∆ 1n < 1/16.Write T −1 = ∆ 1n + (nq) −1 and where A 2 is defined as in Lemma 1.We shall prove, , where δ 0 and ∆ * are defined as in Lemma 3, then where δ N is defined as in Theorem 1.