Mixed Of Martingale Linear Process Functional Central Limit Theorem

In the past two decades, econometrics has achieved significant development in all of its diverse fields. This not only makes econometrics itself an major interdisciplinary subject, but also makes it an important role in modern economics and finance. The development of a rigorous econometric theory strongly rooted in the latest advances in statistical theory is a major improvement in the field. However, it is a pity that most of the classical results are concerned about the i.i.d. situation or martingale case, and there are relatively less literature talking about general dependent situation. In this paper, by applying probability limit theory, we talk about the asymptotic properties for some econometric models and their applications.Chapter 1 introduces the weak convergence and other results of dependent random variables named mixingale.Let {X_n, n≥1} be random variables in (Ω, F,P) . {F_n, n≥1} is F the sub-σsequences, which increase as n increases. For p > 0 , let‖X‖_p = {E|X|^p)^1/p and E_n(X) = E(X|F_n) .Definition 0.1 Let p>1 , {Xn,F,n≥1} will be called L_p -mixingale, if there exist non-negative sequence {c_n} and {μ(m)} ,whereμ(m)â†'0 (mâ†'∞) ,such that for all n≥1 and m≥0 ,‖E_n-mX_n‖)p≤μ(m)c_n,‖X_n - E_n+mX_n‖_p≤μ(m+1)c_n.We will introduce the weak convergence and other classical results of the mixingale in Chapter 1.The functional central limit theory of the linear process plays an important role in time series analysis. A vast amount of literature is devoted to the study of the asymptotics for linear process under various assumptions on the innovations and the coefficients. In Chapter 2, we discuss a linear process defined bywhere {X_t, -∞< t <∞} is a means zero mixingale . {θ_j, j≥0} is a sequence of real numbers that satisfying (?)|θ_j| <∞.We will show that normalized partial sum process of the {Z-t} in model (1) convergences to a Wiener process.In Chapter 3, we show how to apply the functional central limit of the normalizedpartial sum process of the {Z_t} in model (1) in the linear process with a unit root.