© Hindawi Publishing Corp. CAUCHY APPROXIMATION FOR SUMS OF INDEPENDENT RANDOM VARIABLES

We use Stein's method to find a bound for Cauchy approximation. 
The random variables which are considered need to be independent.


Introduction. In Stein's work
, the aim was to show convergence in distribution to the normal.His technique was novel.Stein's technique was free from Fourier methods and relied instead on the elementary differential equation where h : R → R is such that and Nh = E(h(Z)), where Z ∼ N(0, 1).Stein's method was extended from normal distribution to the Poisson distribution by Chen [9].Stein's equation for Poisson with parameter λ is λf (w + 1) − wf (w) = h(w) − P λ h w ∈ Z + , (1.3) where P λ h = E(h(Z)), Z ∼ Poi(λ).Since then, Stein's method has found considerable applications in combinatorics, probability, and statistics.Recent literature pertaining to this method includes Arratia et al. [1,2], Baldi and Rinott [3], Barbour [4,5], Barbour et al. [6], Bolthausen and Götze [7], Chen [10,11], Goldstein and Reinert [12], Goldstein and Rinott [13], Götze [14], and Green [15]; the work of Holst and Janson [16] gives an excellent account of this method.In this paper, we further develop the Stein technique to bound errors for a Cauchy approximation to the distribution of W , the sum of independent random variables.In fact, there are some literatures (e.g., Boonyasombut and Shapiro [8], Neammanee [17], and Shapiro [18]) give a bound of Cauchy approximation in some kind of random variables.But they used Fourier methods.This paper is organized as follows.Main results are stated in Section 2. Proof of main results is in Section 3, while an example is given in Section 4.

Main results.
At the heart of Stein's method lies a Stein equation.For example, are Stein equations for normal and Poisson distribution, respectively.
))dx < ∞}, and for each h ∈ Ᏼ, 2) The Stein equation for Cauchy distribution F It is easy to check that a solution of (2.4) is U h : R → R defined by Fix w 0 ∈ R, and choose h to be the indicator function I (−∞,w 0 ] which is defined by 3), and (2.5), we see that The broad idea of Stein's argument is as follows.First, for any w 0 ∈ R, a function f w 0 : R → R is constructed to solve (2.4) when h is the indicator function I (−∞,w 0 ] .Replacing w by W , for any random variable W , it therefore follows that the difference between P (W ≤ w 0 ) and F(w 0 ) can be expressed as The main results are the following.

.,Y n be identically independent random variables with zero means
Throughout this paper, C stands for an absolute constant with possibly different values in different places.

Proof of main results.
Before we prove the main results, we need the following lemmas.Lemma 3.1.For any real numbers w 0 and w, (1) Proof.(1) follows directly from (2.7).
(2) Before we start the proof, we need the following inequalities: To show (3.1), we define g on (−∞, 0] by g(w) = wF (w).Since g (w) = 2/π (1 + w 2 ) 2 > 0, g is increasing.From this fact and the fact that we have g ≥ 0. Hence, g is increasing and for any w ≤ 0. So (3.1) holds.To show (3.2), we can apply the same argument to the function g on [0, ∞) which is defined by g , it suffices to prove the lemma in the case where w 0 ≥ 0. By (2.7), we have where we have used the fact that 0 ≤ F(w) ≤ 1 in the first inequality and (3.1) and (3.2) in the second inequality.In the case where 0 ≤ w ≤ w 0 , by monotonicity of F and (3.2), we see that Hence, (2) follows from (3.5) and (3.6).
(3) follows immediately from (2) and the fact that (3.7) (4) and ( 5) follow from ( 2) and ( 3) and the facts that wf w 0 (w) (3.8) Lemma 3.2.Let (W , W ) be an exchangeable pair of random variables, that is, for any Borel sets B and B on R, and there exists λ > 0 such that where E W W is the conditional expectation of W with respect to W .Then, for any function f : R → R, for which there exists C > 0 such that for all w ∈ R, Moreover, for any w 0 ∈ R.

Lemma 3.3. Let (W , W ) be an exchangeable pair of random variables such that
with λ > 0.Then, for any w 0 ∈ R, (3.17) Proof.Let w 0 ∈ R. For W < W , we see that and by the same argument we can show that for W < W. So, (3.20) By Lemma 3.2, we have where we have used (3.20) in the last equality.
For fixed w, we define F : R 2 → R by Then, F is antisymmetric.Since W and W are exchangeable, EF (W , W ) = 0. Thus, Proof of Theorem 2.1.Let X 1 ,X 2 ,...,X n be independent random variables and W = X 1 + X 2 +•••+X n .In order to prove the theorem, we introduce additional random variables I, X 1 , X 2 ,..., X n , and W defined in the following way.The random variables I, X 1 ,X 2 ,...,X n , X 1 , X 2 ,..., X n are independent, I is uniformly distributed over the index set {1, 2,...,n}, each X i has the same distribution as the corresponding X i and W = W + ( X I − X I ).Then, (W , W ) is an exchangeable pair.We note that Then, the assumptions of Lemma 3.3 are satisfied with λ = 1/n.Moreover, we know that To prove the theorem, let w 0 ∈ R. By Lemma 3.3, we obtain + n sup w∈R f w 0 (w) + n sup w∈R f w 0 (w) where the fourth inequality comes from (4) and (5) of Lemma 3.1 and the last inequality comes from (3.25).Since X i and X i are independent and have the same distribution, Hence, (3.29) Next, we will give a bound of 2nE( This completes the proof.

Proof of Corollary 2.2. Using
Taylor's formula, we see that Hence, which implies that Clearly, that Hence, by (4.3) and (4.4), the example is proved.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation