### 博碩士論文 80221014 詳細資訊

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(Asymptotic properties of nonlinear moving averages)

 ★ 一些關於雙獨立序列的機率收斂定理

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(i). \$K(X^-_n,X^+_n)\$, (ii). H(X_n),

Giraitis and Surgailis 1999 的文章中, 證明了關於非因果性長記憶過程 \$X_n\$(non-causal)之經驗過程(empirical process)得一個非\$sqrt{N}\$的極限定理. 他們假設\$varepsilon_1\$的特徵函數滿足一些光滑的條件, 並且也需要相當高階的動差條件.

Part I. Let \$X_n=sum_{j=1}^{infty}a_jvarepsilon_{n-j}\$, where \$(varepsilon_i)_{i=-infty}^{infty}\$ are iid with mean 0 and finite second moment and the \$a_i\$ are assumed \$|a_i|=O(i^{- eta})\$ with \$ eta>frac12\$. For a large class of Borel measurable functions \$K(x,y)\$, the Berry-Esseen type rate of convergence for \$Q_N/sqrt{N}\$, \$Q_N=sum_{n=1}^N[K(X_{n+t_1},X_{n+t_2})-EK(X_{n+t_1},X_{n+t_2})], t_1 By fully exploring the linear structure of \$X_n\$, a new type of finite distribution-free orthogonal expansion is develope so that \$Q_N\$ can very well approximated by some random quantities which have nice structures and can be handled with less difficulty. At th end, we give two related examples as applications. One is concerning the zero crossing numbers for a Gaussian long-memory linear process and the other is about the frequency of a given pattern appearing in a random
binary expansion.
Part II. Consider the random vector \$(X^-_n,X^+_n)\$, where \$X^-_n=sum_{j=1}^{infty}a_jvarepsilon_{n-j}\$ and
\$X^+_n=sum_{j=1}^{n+j-1}\$, where \$(a_j)_{jge 1},(b_j)_{jge 1}\$ are two sequences of real number satisfying \$sum_{j=1}^{infty}(a^2_j+b^2_j)

★ 收斂速率
★ 漸近性質
★ 非因果移動平均
★ 非同時泛函
★ 短記憶過程
★ 長記憶過程
★ 非中央極限定理

★ rate of convergence
★ asymptotic properties
★ noncausal moving averages
★ noninstantaneous functional
★ short-memory process
★ long-memory process
★ non-central limit theorem

Contents
Abstract
Part I. The rate of convergence for noninstantaneous functional of linear rocesses
1 Preliminaries
2 Main results
3 Proof
4 Technical lemmas
5 Zero-crossing countings
Part II. The limit theorems for nonlinear functional of noncausal moving averages
1 Introduction
2 Notation and Preliminaries
3 Main Results
4 Proofs of main results
5 Technical Lemmas
References