| 摘要: | 本論文共分兩部分:
第一部份. 線性過程之非同時泛函的收斂速率
令$X_n=sum_{i=1}^{infty}a_ivarepsilon_{n-i}$, 其中 $(varepsilon_i)_{i=-infty}^{infty}$ 為 iid, $Evarepsilon_1=0, Evarepsilon_1^2<infty$, 且$a_i$滿足$|a_i|=O(i^{- eta}), eta>frac12$. 設 $K(x,y)$為Borel 可測函數. 這部分主要證明當$Q_N/sqrt{N}$滿足中央極限定理而其極限變異數為正數時, 它的Berry-Esseen 收斂速率. ($Q_N=sum_{n=1}^N [K(X_{n+t_1},X_{n+t_2})-EK(X_{n+t_1},X_{n+t_2})], t_1 藉由對$X_n$的線性結構的掌握, 除了動差(moment)的假設外, 對於$X_n$的分布函數並沒有其他假設, 我們期待$X_n$的非線性泛函能有一種正交展開式. 藉由對$K(X_{n+t_1},X_{n+t_2})$的展開式, 以及$ell$-逼近法, 我們可以使用Stein 的分區法(blocking), 將 $Q_N/sqrt{N}$的Berry-Esseen收斂速率求出來.
我們列出兩個重要的例子作為應用, 其中之一為Gaussian長記憶過程的過零數(zero-crossing)的漸進性質, 另外一個是關於Non-Gaussian短記憶過程隨機二元展開式句型頻率的極限問題.
第二部分. 非因果性移動平均的非線性轉換的極限定理
這部分主要在研究以下兩種非線性移動平均的漸進性質:
(i). $K(X^-_n,X^+_n)$, (ii). H(X_n),
其中$X^-_n=sum_{j=1}^{infty}a_jvarepsilon_{n-j}, X^+_n=sum_{j=1}^{infty}b_jvarepsilon_{n+j-1}$, 且 $X_n=X^-_n+X^+_n$, 而 $(a_j)_{jge 1}, (b_j)_{jge 1}$ 為滿足$sum_{j=1}^{infty}(a^2_j+b^2_j)<infty, a_0=0,b_0=0$ 的兩實數列. $K:Re^2o Re$ 及 $H:ReoRe$為 Borel 可測函數, 滿足 $E[K(X^-_n,X^+_n)]^2<infty,EH^2(X_n)<infty$. $(varepsilon_i)_{i=-infty}^{infty}$, 為 iid, $Evarepsilon_1=0,Evarepsilon^2_1<infty$.
不論是(i)或(ii), 所考慮的都是非因果性的移動平均, 也就是說它們的可測性包含過去和未來.
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.
經由改良 Ho and Hsing 1997的望遠鏡法, 我們不僅降低 Giraitis and Surgailis 1999的動差條件, 我們也得到較高街的展開式和極限定理. 我們稱這個改良的方法為 '雙重$ell$-近似法.'
;The thesis is divided into two parts.
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 $sum_{j=1}^{infty}(a^2_j+b^2_j)<infty$, $a_0=b_0=0$. $(varepsilon_i)_{i=-infty}^{infty}$ are iid with mean 0 and finite second moment . We study the limit properties of the partial sums $sumlimits_{n=1}^N K(X^-_n,X^+_n)$ of non-causal nonlinear moving averages $K(X^-_n,X^+_n)$. The original techniques used by Ho and Hsing (1997) are not directly applicable. The difficulties stems from the non-causality which destroys the orthogonality of their decompostions. A double $ell$-approxiamtion device propose to remedy the problem. |
