Weak Dependent Coefficients And Technical Analysis

Time series and random fields are main topics in modern statistical techniques. This is because they are essential for applications where randomness plays an important role. To describe the asymptotic behavior of certain time series or random fields,many ways of modeling the weak dependence have already been work out. One of the most popular is the notion of mixing,there exists a wide literature on limit theorems and statistics under various classical mixing conditions such as strong mixing condition(α-mixing),absolute regularity (β-mixing), orφ-mixing. However, many commonly used models for real-world phenomena do not satisfy classical mixing conditions. Moreover, a main inconvenience of mixing assumptions is the difficulty of checking them because the calculation involves the complicated manipulation of taking the supremum over twoσ- algebras.Some alternative ways have been proposed to overcome the disadvantages of strong mixing conditions.These coefficients can be expressed in terms of covariance or conditional expectations of some functions of random variables, the main advantage is that these dependent conditions contain lots of pertinent examples and can be used to deriving limit theorems and statistical applications.In this work, we shall provide another look at the fundamental issue of dependence. Our primary goal is to introduce a previously undescribed type of dependence measures under integral probability metric that are quite different from strong mixing conditions. In this frame we obtain a new covariance inequality and under conditions with quite simple forms, we present limit theorems for partial sums and empirical processes.The dissertation contains five chapters. It is organized in the following way.Chapter 1 reviews some conceptions of the existing weak dependence coefficients and recent developments of concerned problems,introduce the definitions and the applied rules in guiding bargain of several technical analysis indexes (Bollinger band,RSI,Roc)which are needed in the following chapter.Finally, we put forward the main research results in this dissertation.Chapter 2 investigates the properties of corresponding statistics on some popular technical analysis indexes for AR- ARCH model as real stock market . Under the given conditions, we show that these processes are asymptotically stationary and the law of large numbers hold for frequencies of the stock prices falling out normal scope of these technical analysis indexes under AR-ARCH by using mixing conditions, and give the rate of convergence in the case of nonstationary initial values, which give a mathematical rationale for these methods of technical analysis in supervising the security trends.The purpose of Chapter 3 is to propose a simple weak dependence coefficient(Γ-weak dependence) between two random variables under the bounded Lipschitz metric. We show that our coefficient may be also used to obtain covariance inequality similar to Rio(1993).But our codition is much easier to compute for some financial time series models, moreover, we can give the computable rate of convergence for the dependence coefficient.Chapter 4 proves the strong law of large numbers,Glivenko-Cantelli Lemma underΓ-weak dependence, and we can further investigate moment inequalities for associated r.v.s.in terms of this dependent coefficient.Chapter 5 introduce a new dependence coefficient (γ-weak dependence) under the wasserstein metric according to asymptotic independent between finite dimensional joint distribution and its marginal distributions' product for random variables ,and some examples are given. We obtain central limit theorem of empirical process underγ-weak dependence following from the tightness criterion given in Theorem 2.1 of Shao and Yu(1996).