Bounds on Normal Approximation on a Half Plane in Multidimension

respectively, where C0 and C1 are positive constants. The uniform version was independently discovered by (Berry, 1941, p. 122-136) and (Esseen, 1945, p. 1-125) and the non-uniform version was discovered by (Nagaev, 1965, p. 214-235). Without assuming the identically of Xi, the best constant C0 and C1 were given by (Shevtsova, 2010, p. 862-864) and (Paditz, 1989, p. 453-464), respectively. The results are as follows: Theorem 1.1 (Shevtsova, 2010, p. 862-864) Let Xi, 1 ≤ i ≤ n, be independent random variables such that EXi = 0 and E|Xi| < ∞. Assume that n ∑


Introduction
For n ∈ N, let X i , 1 ≤ i ≤ n be independent and identically distributed random variables with zero means and and Φ 1 the standard normal distribution in R. Suppose that E|X i | 3 < ∞ for 1 ≤ i ≤ n.The uniform and non-uniform versions of the Berry-Esseen inequality are and respectively, where C 0 and C 1 are positive constants.The uniform version was independently discovered by (Berry, 1941, p. 122-136) and (Esseen, 1945, p. 1-125) and the non-uniform version was discovered by (Nagaev, 1965, p. 214-235).Without assuming the identically of X i , the best constant C 0 and C 1 were given by (Shevtsova, 2010, p. 862-864) and (Paditz, 1989, p. 453-464), respectively.The results are as follows: Theorem 1.1 (Shevtsova, 2010, p. 862-864) Let X i , 1 ≤ i ≤ n, be independent random variables such that EX i = 0 and Theorem 1.2 (Paditz, 1989, p. 453-464) Under the assumptions of theorem 1.1, we have for all real numbers x. (Chen, 2001, p. 236-254) relaxed the condition to the finiteness of the second moments and gave uniform and non-uniform versions of the inequality.The constant of the non-uniform version was given by (Neammanee, 2007, p. 1-10).Here are the results.
Theorem 1.3 (Chen, 2001, p. 236-254) and for all real numbers x, Theorem 1.4 (Neammanee, 2007, p. 1-10) Under the assumptions of theorem 1.3, we have The reduction they make is truncation.This method make the random variables become bounded random varibles.In the case that each X i is bounded, the uniform and non-uniform versions were given in (Chen, 2005, p. 1-59) and (Chaidee, 2005), respectively.
Theorem 1.5 (Chen, 2005, p. 1-59) Theorem 1.6 (Chaidee, 2005) Under the assumptions of theorem 1.5, there exists a constant C which does not depend on δ 0 such that for every real numbers x, For multidimensional case, let k ∈ N and Y i = (Y i1 , Y i2 , . . ., Y ik ) be independent and identically distributed random vectors in R k with zero means and covariance identity matrices I k .Define Let F n be the distribution function of W n and Φ k the standard Gaussian distribution in R k .(Bergström, 1945, p. 106-127) guaranteed that F n converges weakly to Φ k for large n.The uniform and non-uniform bounds of this convergence have been repeatedly refined over subsequent decades by many researchers such as (Esseen, 1945, p. 1-125), (Rao, 1961, p. 359-361) , (Bahr, 1967, p. 61-69), (Bahr, 1967, p. 71-88) and (Bhattacharya, 1970, p. 68-86), etc.For the assumption that Esseen, 1945, p. 1-125) gave a uniform bound on this convergence which is of the form C is an absolute constant depending only on the moment.(Rao, 1961, p. 359-361)  k+1) .
(1) (Bahr, 1967, p. 71-88) assumed for an integer s > k > 1 and improved the rate of convergence in (1) by the inequality In the case that each Y i may not be identically distributed random vectors, (Bhattacharya, 1970, p. 68-86) and gave a bound of the approximation as in (2) on any Borel subset of R k .For a non-uniform version, (Bahr, 1967, p. 61-69) is the first one who investigated this version.He assumed the identically assumption on each Y i and gave the rate of convergence on B k (r).Under the finiteness assumption of the s th moments, for integer s ≥ 3, the result is where m is the largest eigenvalue of the covariance matrix of Y i , d(n) is bounded by one and lim n→∞ d(n) = 0.The aim of this paper is to find bounds on normal approximation to the distribution of W n over the set In this work, assume only that = 1 and give our results on various assumptions, the random variables Our results are as follows: and there exists a constant C which does not depend on δ 0 such that for every real numbers x, and there exists an absulute constant C such that for x ∈ R, Theorem 1.9 and for all real numbers x The proofs of our main theorems are given in the next section.

Proof of Main Theorems
In the proofs of main theorems, we use the Berry-Esseen theorems in R in which the limit distribution is Φ 1 .However, the limit distribution in our theorems is the standard Gaussian distribution Φ k in R k .In the following proposition, we give a relation between Φ 1 and Φ k .
Proposition 2.1 For k ∈ N and x ∈ R, we have ).
Proof of theorem 1.7 Proof: For each 1 ≤ i ≤ n and , 1 ≤ j ≤ k, we define Thus T 1n , T 2n , . . ., T nn are independent, Since Y i has zero mean and covariance matrix I k , By applying theorem 1.5, proposition 2.1 and ( 4)-( 6), we have For the second part, we apply theorem 1.6.The result is for all real numbers x.
Remark The above theorems include the case that each Y i has an indicator covariance matrix I k .