﻿ α-稳定模型驱动的线性自吸引扩散过程中参数的最小二乘估计

# α-稳定模型驱动的线性自吸引扩散过程中参数的最小二乘估计Least Squares Estimation for the Linear Self-Attracting Diffusion Driven by α-Stable Motions

Abstract: Let Ma be an α-stable motion of dimension one with . In this paper, we consider the self-attracting diffusion of the forms where θ>0 and v∈R are two unknown parameters. The main object of this paper is to study the least squares estimation of θ and ν under the discrete observation and discuss the consistency and asymptotic distributions of the two estimators.

1. 引言

${X}_{t}={X}_{0}+{B}_{t}+{\int }_{0}^{t}\text{d}s{\int }_{0}^{s}f\left({X}_{s}-{X}_{u}\right)\text{d}u.$ (1.1)

${X}_{t}={X}_{0}+{B}_{t}+{\int }_{0}^{t}\text{d}s{\int }_{ℝ}f\left(-x\right){\mathcal{L}}^{X}\left(s,{X}_{s}+x\right)\text{d}x,$

$\text{d}{X}_{t}=\sqrt{2}\text{d}{B}_{t}-\left(\frac{1}{t}{\int }_{0}^{t}\nabla W\left({X}_{t}-{X}_{s}\right)\text{d}s\right)\text{d}t,$

${X}_{t}^{\alpha }={M}_{t}^{\alpha }-\theta {\int }_{0}^{t}{\int }_{0}^{s}\left({X}_{s}^{\alpha }-{X}_{u}^{\alpha }\right)\text{d}u\text{d}s+vt,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0,$ (1.2)

${Y}_{t}={\int }_{0}^{t}\left({X}_{t}-{X}_{s}\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 0,$

${X}_{t}={M}_{t}-\theta {\int }_{0}^{t}{Y}_{s}\text{d}s+vt,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 0.$

${X}_{t}={M}_{t}+{\int }_{0}^{t}\text{d}s{\int }_{0}^{s}\text{d}u\text{ }f\left({X}_{s}-{X}_{u}\right).$ (1.3)

2. 预备知识

$\alpha \in \left(0,2\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \in \left(0,\infty \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta \in \left[-1,1\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu \in \left(-\infty ,+\infty \right),$

${\varphi }_{\alpha }\left(u\right)=\left\{\begin{array}{ll}-{\sigma }^{\alpha }{|u|}^{\alpha }\left(1-i\beta \mathrm{sgn}\left(u\right)\mathrm{tan}\frac{\alpha \pi }{2}\right)+i\mu u,\hfill & 当\alpha \ne 1,\hfill \\ -\sigma |u|\left(1+i\beta \frac{2}{\pi }\mathrm{sgn}\left(u\right)\mathrm{log}|u|\right)+i\mu u,\hfill & 当\alpha =1,\hfill \end{array}$

$E{\text{e}}^{iu\eta }={\text{e}}^{{\varphi }_{\alpha }\left(u\right)}.$

$\mu =0$ 时，我们称 $\eta$ 是严格 $\alpha$ -稳定的。此外，若 $\beta =0$，则称 $\eta$ 是对称 $\alpha$ -稳定的。

$\alpha \in \left(0,2\right]$${M}^{\alpha }=\left\{{M}_{t}^{\alpha },t\ge 0\right\}$ 是一个 $\left\{{\mathcal{F}}_{t}\right\}$ -适应过程。如果对任意 $t>s>0$

$\mathbb{E}\left[{\text{e}}^{iu\left({M}_{t}^{\alpha }-{M}_{s}^{\alpha }\right)}|{\mathcal{F}}_{s}\right]={\text{e}}^{\left(t-s\right){\varphi }_{\alpha }\left(u\right)},$

3. θ与v的最小二乘估计量及相合性

${Y}_{{t}_{i}}={\int }_{0}^{{t}_{i}}\left({X}_{{t}_{i}}-{X}_{s}\right)\text{d}s\simeq \underset{k=1}{\overset{i}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\ge 0,$

${X}_{{t}_{i}}={M}_{{t}_{i}}-\theta {\int }_{0}^{{t}_{i}}{Y}_{s}\text{d}s+v{t}_{i}\simeq {M}_{{t}_{i}}-\theta \underset{l=1}{\overset{i}{\sum }}\left[\underset{k=1}{\overset{l-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\right]h+v{t}_{i},$

${X}_{{t}_{i}}-{X}_{{t}_{i-1}}=-\left(\theta {Y}_{{t}_{i-1}}-v\right)h+\left({M}_{{t}_{i}}-{M}_{{t}_{i-1}}\right),$ (3.1)

${\rho }_{n}\left(\theta ,v\right)=\underset{i=1}{\overset{n}{\sum }}{|{X}_{{t}_{i}}-{X}_{{t}_{i-1}}+\left(\theta {Y}_{{t}_{i-1}}-v\right)h|}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0.$

$\begin{array}{c}\stackrel{^}{\theta }=\frac{\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\underset{i=1}{\overset{n}{\sum }}\left({X}_{ih}-{X}_{\left(i-1\right)h}\right)-n\underset{i=1}{\overset{n}{\sum }}\left({X}_{ih}-{X}_{\left(i-1\right)h}\right){Y}_{{t}_{i-1}}}{nh\underset{i=1}{\overset{n}{\sum }}{\left({Y}_{{t}_{i-1}}\right)}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}\\ =\theta -\frac{n\underset{i=1}{\overset{n}{\sum }}\left({M}_{ih}-{M}_{\left(i-1\right)h}\right){Y}_{{t}_{i-1}}}{nh\underset{i=1}{\overset{n}{\sum }}{\left({Y}_{{t}_{i-1}}\right)}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}+\frac{{M}_{nh}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}}{nh\underset{i=1}{\overset{n}{\sum }}{\left({Y}_{{t}_{i-1}}\right)}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}\\ =\theta -\frac{n\underset{i=1}{\overset{n}{\sum }}\left({M}_{ih}-{M}_{\left(i-1\right)h}\right)\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h}{nh\underset{i=1}{\overset{n}{\sum }}{\left(\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\right)}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\right)}^{2}}\\ \text{\hspace{0.17em}}+\frac{\underset{i=1}{\overset{n}{\sum }}\left({M}_{ih}-{M}_{\left(i-1\right)h}\right)\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h}{nh\underset{i=1}{\overset{n}{\sum }}{\left(\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\right)}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\right)}^{2}}.\end{array}$ (3.2)

$\stackrel{^}{v}=v+\frac{1}{n}\left(\stackrel{^}{\theta }-\theta \right)\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{i}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h+\frac{1}{nh}\underset{i=1}{\overset{n}{\sum }}\left({M}_{ih}-{M}_{\left(i-1\right)h}\right).$ (3.3)

$\stackrel{^}{{Y}_{t}}={t}^{\frac{1}{\alpha }-1}{Y}_{t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}F\left(s\right)={t}^{\frac{1}{\alpha }-1}s{\text{e}}^{-\frac{1}{2}\theta \left({t}^{2}-{s}^{2}\right)}.$ (3.4)

${\lambda }_{F}={\left({\int }_{0}^{t}{t}^{1-\alpha }{s}^{\alpha }{\text{e}}^{-\frac{\alpha }{2}\theta \left({t}^{2}-{s}^{2}\right)}\text{d}s\right)}^{\frac{1}{\alpha }}.$

$\underset{k=1}{\overset{i}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\simeq {Y}_{{t}_{i}}={\text{e}}^{-\frac{1}{2}\theta {t}_{i}^{2}}{\int }_{0}^{{t}_{i}}s{\text{e}}^{\frac{1}{2}\theta {s}^{2}}\text{d}{M}_{s}+\frac{v}{\theta }\left(1-{\text{e}}^{\frac{1}{2}\theta {t}_{i}^{2}}\right).$ (3.5)

$\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\to \frac{v}{\theta }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right),$ (3.6)

${X}_{{t}_{i}}-{X}_{{t}_{i-1}}={M}_{{t}_{i}}-{M}_{{t}_{i-1}}-\theta h{Y}_{{t}_{i-1}}+vh,$

$\theta h\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}=\underset{i=1}{\overset{n}{\sum }}\left[-\left({X}_{{t}_{i}}-{X}_{{t}_{i-1}}\right)+\left({M}_{{t}_{i}}-{M}_{{t}_{i-1}}\right)\right]+nvh.$

$\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}=-\frac{{X}_{t}}{nh\theta }+\frac{{M}_{t}}{nh\theta }+\frac{v}{\theta }.$

$|\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}-\frac{v}{\theta }|=\frac{1}{\theta }|\frac{{X}_{{t}_{n}}}{nh}-\frac{{M}_{{t}_{n}}}{nh}|.$

${X}_{T}\to {X}_{\infty }={\int }_{0}^{\infty }h\left(s\right)\text{d}{M}_{s}+v{\int }_{0}^{\infty }h\left(s\right)\text{d}s\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right),$

$\underset{n\to +\infty }{\mathrm{lim}}|\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}-\frac{v}{\theta }|=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right).$

$\stackrel{^}{\theta }\to \theta \text{}\left(a.s.\right)$$n\to \infty .$

$\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\left({M}_{{t}_{i}}-{M}_{{t}_{i-1}}\right)={\int }_{0}^{{t}_{n}}{\varphi }_{n}\left(s\right)\text{d}{M}_{s}.$ (3.7)

${\tau }_{n}\left({t}_{n}\right)={\int }_{0}^{{t}_{n}}{|{\varphi }_{n}\left(t\right)|}^{\alpha }\text{d}t$，则有

${\tau }_{n}\left({t}_{n}\right)={\int }_{0}^{{t}_{n}}\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{\alpha }{1}_{\left({t}_{i-1},{t}_{i}\right]}\left(t\right)\text{d}t=\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{\alpha }h.$ (3.8)

$\frac{n\underset{i=1}{\overset{n}{\sum }}\left({M}_{ih}-{M}_{\left(i-1\right)h}\right)\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h}{nh\underset{i=1}{\overset{n}{\sum }}{\left(\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\right)}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{i-1}{\sum }}\left({X}_{ih}-{X}_{\left(k-1\right)h}\right)h\right)}^{2}}=\frac{{\int }_{0}^{{t}_{n}}{\varphi }_{n}\left(t\right)\text{d}{M}_{t}}{{\tau }_{n}\left({t}_{n}\right)}\cdot \frac{n{\tau }_{n}\left({t}_{n}\right)}{nh\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}.$ (3.9)

$\theta >0$，则 $\stackrel{^}{{Y}_{t}}$ 是遍历的，且 $t\to \infty$$\stackrel{^}{{Y}_{t}}⇒\stackrel{^}{{Y}_{\infty }}={S}_{\alpha }\left({\left(\alpha \theta \right)}^{-\frac{1}{\alpha }},0,0\right)$。因此由遍历定理得

$\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{|\stackrel{^}{{Y}_{{t}_{i-1}}}|}^{\alpha }=\mathbb{E}\left[{\stackrel{^}{{Y}_{\infty }}}^{\alpha }\right]=\infty \text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right).$

$\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{\alpha }=\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{\left({t}_{i}\right)}^{\alpha -1}{|\stackrel{^}{{Y}_{{t}_{i-1}}}|}^{\alpha }\to \infty \text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right),$

$\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{2}=\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{t}_{i}{|\stackrel{^}{{Y}_{{t}_{i-1}}}|}^{2}\to \infty \text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right).$

$\underset{{t}_{n}\to \infty }{\mathrm{lim}}\frac{|{\int }_{0}^{{t}_{n}}{\varphi }_{n}\left(t\right)\text{d}{M}_{t}|}{{\tau }_{n}\left({t}_{n}\right)}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right).$ (3.10)

$\begin{array}{c}0\le \underset{n\to \infty }{\mathrm{lim}}\frac{n{\tau }_{n}\left({t}_{n}\right)}{nh\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}=\underset{n\to \infty }{\mathrm{lim}}\frac{n\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{\alpha }}{n\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}^{2}-{\left(\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{\alpha }}{\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{2}}\le \underset{n\to \infty }{\mathrm{lim}}\frac{{\left(\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{2}\right)}^{\alpha /2}{n}^{\left(2-\alpha \right)/2}}{\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{2}}\le \underset{n\to \infty }{\mathrm{lim}}{\left(\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{|{Y}_{{t}_{i-1}}|}^{2}\right)}^{-\frac{2-\alpha }{2}},\end{array}$ (3.11)

$\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{|\stackrel{^}{{Y}_{{t}_{i-1}}}|}^{2}=\mathbb{E}\left[{\stackrel{^}{{Y}_{\infty }}}^{2}\right]=\infty \text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(a.s.\right).$

$\frac{{M}_{nh}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}}{nh\underset{i=1}{\overset{n}{\sum }}{\left({Y}_{{t}_{i-1}}\right)}^{2}-h{\left(\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}=\frac{\frac{{M}_{nh}}{nh}\cdot \frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}}{\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{\left({Y}_{{t}_{i-1}}\right)}^{2}-{\left(\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}}$ (3.12)

$n\to \infty$ 时几乎必然收敛于0。综上，命题3.1成立。

$\stackrel{^}{v}\to v\text{}\left(a.s.\right)$$n\to \infty .$

4. $\stackrel{^}{\theta }$$\stackrel{^}{v}$ 的渐近分布

$\alpha$ -稳定过程驱动的O-U过程的参数估计的渐近分布 [21]，我们自然地考虑到有类似的结果。

$n{\left(\frac{\mathrm{log}n}{n}\right)}^{-\frac{1}{\alpha }}{h}^{2}\left(\stackrel{^}{\theta }-\theta \right)⇒\frac{{Z}_{1}}{2{Z}_{0}},$

${Y}_{{t}_{i}}={\text{e}}^{-\frac{1}{2}\theta {t}_{i}^{2}}\underset{k=1}{\overset{i}{\sum }}{\int }_{{t}_{k-1}}^{{t}_{k}}s{\text{e}}^{\frac{1}{2}\theta {s}^{2}}\text{d}{M}_{s}+\frac{v}{\theta }\left(1-{\text{e}}^{-\frac{1}{2}\theta {t}_{i}^{2}}\right).$

${V}_{k-1}={\int }_{{t}_{k-1}}^{{t}_{k}}s{\text{e}}^{\frac{1}{2}\theta {s}^{2}}\text{d}{M}_{s}$，由 $\alpha$ -稳定随机积分的内部时性质可知 ${V}_{k-1}$${M}_{{\tau }_{k-1}}$ 同分布，其中

${\tau }_{k-1}={\int }_{{t}_{k-1}}^{{t}_{k}}{|s{\text{e}}^{\frac{1}{2}\theta {s}^{2}}|}^{\alpha }\text{d}s={t}_{k}^{\alpha }{\text{e}}^{\frac{\alpha }{2}\theta {t}_{k}^{2}}h.$

${U}_{k-1}={V}_{k-1}/{\tau }_{k-1}^{\frac{1}{\alpha }}$，由稳定分布的收缩性质可知 ${U}_{0},{U}_{1},{U}_{2},\cdots$ 是独立同分布的随机变量且具有稳定分布 ${S}_{\alpha }\left(1,0,0\right)$。设 ${c}_{i,h}={\text{e}}^{-\frac{1}{2}\theta {t}_{i}^{2}}$，则 ${Y}_{i}$ ( ${Y}_{{t}_{i}}$ 简写为 ${Y}_{i}$ )可以表示为

${Y}_{i}={h}^{\frac{1}{\alpha }+1}{c}_{i,h}\underset{k=1}{\overset{i}{\sum }}k\frac{1}{{c}_{k,h}}{U}_{k-1}+\frac{v}{\theta }\left(1-{\text{e}}^{-\frac{1}{2}\theta {t}_{i}^{2}}\right),$

${M}_{{t}_{i+1}}-{M}_{{t}_{i}}$ 可以表示为 ${h}^{\frac{1}{\alpha }}{U}_{i}$

$\underset{x\to \infty }{\mathrm{lim}}{x}^{\alpha }P\left({U}_{1}>x\right)={C}_{\alpha }/2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\to \infty }{\mathrm{lim}}{x}^{\alpha }P\left({U}_{1}<-x\right)={C}_{\alpha }/2.$

${a}_{n}=inf\left\{x:P\left(|{U}_{1}|>x\right)\le {n}^{-1}\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{˜}{{a}_{n}}=inf\left\{x:P\left(|{U}_{0}{U}_{1}|>x\right)\le {n}^{-1}\right\}.$

${a}_{n}={\left({C}_{\alpha }n\right)}^{\frac{1}{\alpha }},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{˜}{{a}_{n}}={C}_{\alpha }^{\frac{2}{\alpha }}{\left(n\mathrm{log}n\right)}^{\frac{1}{\alpha }}.$

$\left({a}_{n}^{-2}\underset{i=1}{\overset{n}{\sum }}{U}_{i}^{2},\text{}{\stackrel{˜}{{a}_{n}}}^{-1}\underset{i=1}{\overset{n}{\sum }}{U}_{i}{U}_{i+1},\cdots ,{\stackrel{˜}{{a}_{n}}}^{-1}\underset{i=1}{\overset{n}{\sum }}{U}_{i}{U}_{i+m}\right)⇒\left({Z}_{0},{Z}_{1},\cdots ,{Z}_{m}\right),$

$\begin{array}{c}n{\left(\frac{\mathrm{log}n}{n}\right)}^{-\frac{1}{\alpha }}{h}^{2}\left(\stackrel{^}{\theta }-\theta \right)=-\frac{{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}\left({M}_{{t}_{i+1}}-{M}_{{t}_{i}}\right){Y}_{{t}_{i}}}{{n}^{-3-\frac{2}{\alpha }}{h}^{-3-\frac{2}{\alpha }}\left[nh\underset{i=1}{\overset{n-1}{\sum }}{\left({Y}_{i}\right)}^{2}-h{\left(\underset{i=1}{\overset{n-1}{\sum }}{Y}_{i}\right)}^{2}\right]}+\frac{{n}^{-2}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{2}{\alpha }}{M}_{nh}\underset{i=1}{\overset{n-1}{\sum }}{Y}_{{t}_{i}}}{{n}^{-3-\frac{2}{\alpha }}{h}^{-3-\frac{2}{\alpha }}\left[nh\underset{i=1}{\overset{n-1}{\sum }}{\left({Y}_{i}\right)}^{2}-h{\left(\underset{i=1}{\overset{n-1}{\sum }}{Y}_{i}\right)}^{2}\right]}\\ =-\frac{{\Phi }_{1}\left(n\right)}{{\Phi }_{3}\left(n\right)}+\frac{{\Phi }_{2}\left(n\right)}{{\Phi }_{3}\left(n\right)}.\end{array}$ (4.1)

${\Phi }_{3}\left(n\right)-{n}^{-2-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{k=1}{\overset{i}{\sum }}\frac{1}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}{\to }_{p}0.$

$\begin{array}{c}{\Phi }_{3}\left(n\right)={n}^{-2-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}{\left[\underset{k=1}{\overset{i}{\sum }}\frac{k}{{c}_{k,h}}{U}_{k-1}\right]}^{2}+{n}^{-2-\frac{2}{\alpha }}{h}^{-2-\frac{2}{\alpha }}\frac{{v}^{2}}{{\theta }^{2}}\underset{i=1}{\overset{n-1}{\sum }}{\left(1-{c}_{i,h}\right)}^{2}\\ \text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}+{n}^{-2-\frac{2}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\frac{2v}{\theta }\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}\left(1-{c}_{i,h}\right)\underset{m=1}{\overset{i}{\sum }}\frac{m}{{c}_{m,h}}{U}_{m-1}-{n}^{-1-\frac{2}{\alpha }}{h}^{-2-\frac{2}{\alpha }}{\left(\frac{1}{n}\underset{i=1}{\overset{n-1}{\sum }}{Y}_{{t}_{i-1}}\right)}^{2}\\ :={\Sigma }_{0}+{\Sigma }_{1}+{\Sigma }_{2}+{\Sigma }_{3}.\end{array}$ (4.2)

$\begin{array}{c}P\left(|{\Sigma }_{2}|>\epsilon \right)\le {\epsilon }^{-1}\mathbb{E}|{U}_{m-1}|\frac{2v}{\theta }{n}^{-2-\frac{2}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}\left[\left(1-{c}_{i,h}\right){c}_{i,h}\left(\frac{i}{{c}_{i,h}}+\underset{m=1}{\overset{i-1}{\sum }}\frac{m}{{c}_{m,h}}\right)\right]\\ \le C{n}^{-2-\frac{2}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}\left[i-i{c}_{i,h}+\frac{{\left(i-1\right)}^{2}{c}_{i,h}}{{c}_{i-1,h}}-\frac{{\left(i-1\right)}^{2}{c}_{i,h}^{2}}{{c}_{i-1,h}}\right]\\ \le C{n}^{-\frac{2}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\end{array}$ (4.3)

$\begin{array}{c}{\Sigma }_{0}={n}^{-2-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}{\left[\underset{k=1}{\overset{i}{\sum }}\frac{k}{{c}_{k,h}}{U}_{k-1}\right]}^{2}\\ ={n}^{-2-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{k=1}{\overset{i}{\sum }}\frac{{k}^{2}}{{c}_{k,h}^{2}}{U}_{k-1}^{2}+{n}^{-2-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{p=1}{\overset{i}{\sum }}\underset{q=1,q\ne p}{\overset{i}{\sum }}\frac{pq}{{c}_{p,h}{c}_{q,h}}{U}_{p-1}{U}_{q-1}\\ :={\Sigma }_{01}+{\Sigma }_{02}.\end{array}$ (4.4)

$\begin{array}{l}P\left({n}^{-2-\frac{2}{\alpha }}|\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{p=1}{\overset{i}{\sum }}\underset{q=1,q\ne p}{\overset{i}{\sum }}\frac{pq}{{c}_{p,h}{c}_{q,h}}{U}_{p-1}{U}_{q-1}|>\epsilon \right)\\ \le P\left({n}^{-2-\frac{2}{\alpha }}|\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{p=1}{\overset{i}{\sum }}\underset{q=1,q\ne p}{\overset{i}{\sum }}\frac{pq{U}_{p-1}{U}_{q-1}}{{c}_{p,h}{c}_{q,h}}{1}_{\left(|{U}_{p-1}{U}_{q-1}|\le \stackrel{˜}{{a}_{n}}\right)}|>\frac{\epsilon }{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+P\left({n}^{-2-\frac{2}{\alpha }}|\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{p=1}{\overset{i}{\sum }}\underset{q=1,q\ne p}{\overset{i}{\sum }}\frac{pq{U}_{p-1}{U}_{q-1}}{{c}_{p,h}{c}_{q,h}}{1}_{\left(|{U}_{p-1}{U}_{q-1}|>\stackrel{˜}{{a}_{n}}\right)}|>\frac{\epsilon }{2}\right)\\ :={B}_{1}+{B}_{2}.\end{array}$ (4.5)

$D\left(U\right)={U}_{p-1}{U}_{q-1},\text{\hspace{0.17em}}{D}^{\prime }\left(U\right)={U}_{{p}^{\prime }-1}{U}_{{q}^{\prime }-1}$。由切比雪夫不等式知，

$\begin{array}{c}{B}_{1}\le C{\left(\epsilon /2\right)}^{-2}{n}^{-4-\frac{4}{\alpha }}\mathbb{E}{\left[\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{p=1}{\overset{i}{\sum }}\underset{q=1,q\ne p}{\overset{i}{\sum }}\frac{pq}{{c}_{p-1,h}{c}_{q-1,h}}D\left(U\right){1}_{\left(|D\left(U\right)|\le \stackrel{˜}{{a}_{n}}\right)}\right]}^{2}\\ \le C{\left(\epsilon /2\right)}^{-2}{n}^{-4-\frac{4}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}\underset{r=1}{\overset{n-1}{\sum }}\underset{p,q=1p\ne q}{\overset{i}{\sum }}\underset{{p}^{\prime },{q}^{\prime }=1{p}^{\prime }\ne {q}^{\prime }}{\overset{r}{\sum }}\frac{pq{p}^{\prime }{q}^{\prime }{c}_{i,h}^{2}{c}_{r,h}^{2}}{{c}_{p,h}{c}_{q,h}{c}_{{p}^{\prime },h}{c}_{{q}^{\prime },h}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\mathbb{E}\left[D\left(U\right){1}_{\left(|D\left(U\right)|\le \stackrel{˜}{{a}_{n}}\right)}{D}^{\prime }\left(U\right){1}_{\left(|{D}^{\prime }\left(U\right)|\le \stackrel{˜}{{a}_{n}}\right)}\right].\end{array}$ (4.6)

$\begin{array}{c}{B}_{1}\le 16C{\epsilon }^{-2}{n}^{-4-\frac{4}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}\underset{p=1}{\overset{i}{\sum }}\underset{q=1,q\ne p}{\overset{i}{\sum }}\underset{r=1}{\overset{n-1}{\sum }}\underset{{p}^{\prime }=1}{\overset{r}{\sum }}\underset{{q}^{\prime }=1,{q}^{\prime }\ne {p}^{\prime }}{\overset{r}{\sum }}\frac{pq{p}^{\prime }{q}^{\prime }{c}_{i,h}^{2}{c}_{r,h}^{2}}{{c}_{p,h}{c}_{q,h}{c}_{{p}^{\prime },h}{c}_{{q}^{\prime },h}}{\sigma }_{n}^{2}\\ \le 16C{\epsilon }^{-2}{n}^{-4-\frac{4}{\alpha }}{\left[\underset{i=1}{\overset{n-1}{\sum }}\underset{p=1}{\overset{i}{\sum }}\underset{p=1,q\ne p}{\overset{i}{\sum }}\frac{pq{c}_{i,h}^{2}}{{c}_{p,h}{c}_{q,h}}\right]}^{2}n{\sigma }_{n}^{2}\\ \le 16C{\epsilon }^{-2}{n}^{-4-\frac{4}{\alpha }}{\left[\underset{i=1}{\overset{n-1}{\sum }}{i}^{2}{\left(i-1\right)}^{2}{\text{e}}^{-\theta \left(i-\frac{1}{2}\right){h}^{2}}\right]}^{2}\cdot n{\sigma }_{n}^{2}\\ ={C}^{\prime }{\epsilon }^{-2}{n}^{-4-\frac{4}{\alpha }}{\left(n\mathrm{log}n\right)}^{\frac{2}{\alpha }}\left[{\stackrel{˜}{{a}_{n}}}^{-2}n{\sigma }_{n}^{2}\right].\end{array}$ (4.7)

$\begin{array}{c}{B}_{2}\le 2{\epsilon }^{-1}{n}^{-2-\frac{2}{\alpha }}\mathbb{E}|\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{p=1}{\overset{i}{\sum }}\underset{q=1,q\ne p}{\overset{i}{\sum }}\frac{pq{U}_{p-1}{U}_{q-1}}{{c}_{p,h}{c}_{q,h}}{1}_{\left(|D\left(U\right)|>\stackrel{˜}{{a}_{n}}\right)}\\ \le 2{\epsilon }^{-1}{n}^{-2-\frac{2}{\alpha }}\left[\underset{i=1}{\overset{n-1}{\sum }}{i}^{2}{\left(i-1\right)}^{2}{\text{e}}^{-\theta \left(i-\frac{1}{2}\right){h}^{2}}\right]\mathbb{E}\left[D\left(U\right){1}_{\left(|D\left(U\right)|>\stackrel{˜}{{a}_{n}}\right)}\right]\\ \le C{n}^{-3-\frac{2}{\alpha }}{\left(n\mathrm{log}n\right)}^{\frac{1}{\alpha }}\left[n{\stackrel{˜}{{a}_{n}}}^{-1}\mathbb{E}\left(D\left(U\right){1}_{\left(|D\left(U\right)|>\stackrel{˜}{{a}_{n}}\right)}\right)\right].\end{array}$ (4.8)

${n}^{-2-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{k=1}{\overset{i}{\sum }}\frac{1}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}⇒{C}_{\alpha }^{\frac{2}{\alpha }}{Z}_{0}$

$\begin{array}{c}{\Sigma }_{01}={n}^{-2-\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\underset{k=1}{\overset{i}{\sum }}\frac{1}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}={n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}\frac{1}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}\underset{i=k}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\\ ={n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}{k}^{2}{U}_{k-1}^{2}+{n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}\frac{1}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}\underset{i=k+1}{\overset{n-1}{\sum }}{c}_{i,h}^{2}\\ ={n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}{k}^{2}{U}_{k-1}^{2}+{n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}\frac{{c}_{k+1,h}^{2}}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}-{n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}\frac{{c}_{n,h}^{2}}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\circ \left({n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}\frac{{c}_{k+1,h}^{2}}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}\right)+\circ \left({n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}\frac{{c}_{n,h}^{2}}{{c}_{k,h}^{2}}{k}^{2}{U}_{k-1}^{2}\right)\\ :={D}_{1}+{D}_{2}+{D}_{3}+{D}_{4}+{D}_{5}.\end{array}$ (4.9)

$\delta =\frac{2\alpha }{2+\rho }<\alpha ,\text{\hspace{0.17em}}\rho >0\text{\hspace{0.17em}}\left(\delta /2<\alpha /2<1\right)$，由马尔可夫不等式可知，

$P\left(|{D}_{3}|>\epsilon \right)\le {\epsilon }^{-\frac{\delta }{2}}{n}^{-\delta -\frac{\delta }{\alpha }}{\left(n-1\right)}^{\delta +\frac{\delta }{2}}{\left[{\text{e}}^{-\frac{1}{2}\theta \left(2n-1\right){h}^{2}}\right]}^{\frac{\delta }{2}}\mathbb{E}{|{U}_{k-1}|}^{\delta }.$ (4.10)

$P\left(|{D}_{2}|>\epsilon \right)\le {\epsilon }^{-\frac{\delta }{2}}{n}^{-\delta -\frac{\delta }{\alpha }}{\left[\underset{k=1}{\overset{n-1}{\sum }}{\text{e}}^{-\theta \left(2k-1\right){h}^{2}}\right]}^{\frac{\delta }{2}}\mathbb{E}{|{U}_{k-1}|}^{\delta }.$ (4.11)

$\begin{array}{c}{D}_{1}={n}^{-2-\frac{2}{\alpha }}\underset{k=1}{\overset{n-1}{\sum }}\left(\underset{j=1}{\overset{k}{\sum }}{d}_{j}\right){U}_{k-1}^{2}={n}^{-2-\frac{2}{\alpha }}\underset{j=1}{\overset{n-1}{\sum }}{d}_{j}\underset{k=j}{\overset{n-1}{\sum }}{U}_{k-1}^{2}\\ ={n}^{-2-\frac{2}{\alpha }}{\left(n-1\right)}^{2}\underset{k=0}{\overset{n-1}{\sum }}{U}_{k}^{2}-{n}^{-2-\frac{2}{\alpha }}{\left(n-1\right)}^{2}{U}_{n-1}^{2}-{n}^{-2-\frac{2}{\alpha }}\underset{j=1}{\overset{n-1}{\sum }}{d}_{j}\underset{k=0}{\overset{j-1}{\sum }}{U}_{k}^{2}\\ :={D}_{11}+{D}_{12}+{D}_{13}.\end{array}$ (4.12)

$P\left(|{D}_{12}|>\epsilon \right)\le {\epsilon }^{-\frac{\delta }{2}}{n}^{-\frac{\delta }{\alpha }}\mathbb{E}{|{U}_{n-1}|}^{\delta }$ (4.13)

$P\left(|{D}_{13}|>\epsilon \right)\le {\epsilon }^{-\frac{\delta }{2}}{n}^{-\delta -\frac{\delta }{\alpha }}{\left(\underset{j=1}{\overset{n}{\sum }}{d}_{j}\cdot j\right)}^{\frac{\delta }{2}}\mathbb{E}{|{U}_{n}-1|}^{\delta }\le C{n}^{-\frac{\delta }{\alpha }+\frac{\delta }{2}}$ (4.14)

${\Phi }_{1}\left(n\right)-{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{l}{U}_{l+1}{\to }_{P}0.$

$\begin{array}{c}{\Phi }_{1}\left(n\right)={n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{2}{\alpha }}\left[{h}^{1+\frac{2}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}{U}_{i}\underset{k=1}{\overset{i}{\sum }}\frac{k}{{c}_{k,h}}{U}_{k-1}+{h}^{\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{\Delta }_{{t}_{i}}{U}_{i}\right]\\ ={n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{c}_{i,h}{U}_{i}\underset{k=1}{\overset{i}{\sum }}\frac{k}{{c}_{k,h}}{U}_{k-1}+{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{\Delta }_{{t}_{i}}{U}_{i}\\ ={n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{j=1}{\overset{n-1}{\sum }}\underset{l=0}{\overset{n-1-j}{\sum }}\left(l+1\right)\frac{{c}_{l+j,h}}{{c}_{l+1,h}}{U}_{l}{U}_{l+j}+{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{\Delta }_{{t}_{i}}{U}_{i}\\ ={n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{j=1}{\overset{n-1}{\sum }}\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+j,h}}{{c}_{l+1,h}}{U}_{l}{U}_{l+j}-{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{j=1}{\overset{n-1}{\sum }}\underset{l=n-j}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+j,h}}{{c}_{l+1,h}}{U}_{l}{U}_{l+j}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{\Delta }_{{t}_{i}}{U}_{i}\end{array}$

$\begin{array}{l}={n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{l}{U}_{l+1}+{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{j=2}{\overset{n-1}{\sum }}\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+j,h}}{{c}_{l+1,h}}{U}_{l}{U}_{l+j}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{j=1}{\overset{n-1}{\sum }}\underset{l=n-j}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+j,h}}{{c}_{l+1,h}}{U}_{l}{U}_{l+j}+{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\underset{i=1}{\overset{n-1}{\sum }}{\Delta }_{{t}_{i}}{U}_{i}\\ :={F}_{1}+{F}_{2}+{F}_{3}+{F}_{4}.\end{array}$ (4.15)

$\begin{array}{c}P\left(|{F}_{2}|>\epsilon \right)\le {\epsilon }^{-1}{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\mathbb{E}|\underset{j=2}{\overset{n-1}{\sum }}\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+j,h}}{{c}_{l+1,h}}{U}_{l}{U}_{l+j}|\\ \le {\epsilon }^{-1}{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\mathbb{E}|{U}_{0}|\mathbb{E}|{U}_{1}|\left(n-2\right)\underset{l=1}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+1,h}}{{c}_{l,h}}\\ \le C{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{l=1}{\overset{n-1}{\sum }}\left(l+1\right){\text{e}}^{-\theta \left(l+\frac{1}{2}\right){h}^{2}}\\ \le {C}^{\prime }{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\end{array}$ (4.16)

$\begin{array}{c}{F}_{3}={n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\left[\left(n-1+1\right){U}_{n-1}{U}_{n}\right]+{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{j=2}{\overset{n-1}{\sum }}\underset{l=n-j}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+j,h}}{{c}_{l+1,h}}{U}_{l}{U}_{l+j}\\ :={F}_{31}+{F}_{32}.\end{array}$ (4.17)

$P\left(|{F}_{31}|>\epsilon \right)\le {\epsilon }^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\mathbb{E}|{U}_{n-1}{U}_{n}|\le {\epsilon }^{-1}\mathbb{E}|{U}_{n-1}|\mathbb{E}|{U}_{n}|{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\le C{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}$ (4.18)

$\begin{array}{c}P\left(|{F}_{32}|>\epsilon \right)\le {\epsilon }^{-1}{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\mathbb{E}|{U}_{l}|\mathbb{E}|{U}_{l+j}|\underset{j=2}{\overset{n-1}{\sum }}\underset{l=n-j}{\overset{n-1}{\sum }}\left(l+1\right)\frac{{c}_{l+j}}{{c}_{l+1}}\\ \le C{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{j=2}{\overset{n-1}{\sum }}j\left(n-j+1\right)\frac{{c}_{n}}{{c}_{n-j+1}}\\ \le {C}^{\prime }{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\end{array}$ (4.19)

$P\left(|{F}_{4}|>\epsilon \right)\le {\epsilon }^{-1}{n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{1}{\alpha }}\mathbb{E}|{U}_{i}|\cdot \frac{v}{\theta }\underset{i=1}{\overset{n-1}{\sum }}\left(1-{c}_{i,h}\right)\le C{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{1}{\alpha }}$ (4.20)

${n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{l}{U}_{l+1}⇒\frac{1}{2}{C}_{\alpha }^{\frac{2}{\alpha }}{Z}_{1}.$

${F}_{1}={n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{l}{U}_{l+1}.$ (4.21)

$\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{l}{U}_{l+1}+\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{n-l}{U}_{n-1-l}=\left(n+1\right)\underset{l=0}{\overset{n-1}{\sum }}{U}_{l}{U}_{l+1},$

$\begin{array}{l}P\left(|\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{l}{U}_{l+1}-\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{n-l}{U}_{n-1-l}|>\epsilon \right)\\ \le {\epsilon }^{-1}\mathbb{E}|\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{l}{U}_{l+1}-\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right){U}_{n-l}{U}_{n-1-l}|\\ \le {\epsilon }^{-1}\left[\mathbb{E}|{U}_{l}{U}_{l+1}|-\mathbb{E}|{U}_{n-l}{U}_{n-1-l}|\right]\underset{l=0}{\overset{n-1}{\sum }}\left(l+1\right)=0.\end{array}$ (4.22)

${n}^{-1}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}\left(n+1\right)\underset{l=0}{\overset{n-1}{\sum }}{U}_{l}{U}_{l+1}⇒{C}_{\alpha }^{\frac{2}{\alpha }}{Z}_{1}.$

${\Phi }_{2}\left(n\right)={n}^{-2}{\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-1-\frac{2}{\alpha }}\left[{M}_{nh}\underset{i=1}{\overset{n}{\sum }}{Y}_{i-1}\right]=\frac{{M}_{nh}}{nh}\cdot \frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{Y}_{{t}_{i-1}}\cdot {\left(n\mathrm{log}n\right)}^{-\frac{1}{\alpha }}{h}^{-\frac{2}{\alpha }}$ (4.23)

$n{\left(\frac{\mathrm{log}n}{n}\right)}^{-\frac{1}{\alpha }}{h}^{2}\left(\stackrel{^}{v}-v-\frac{{M}_{nh}}{nh}\right)=\left[\frac{1}{n}\underset{i=1}{\overset{n-1}{\sum }}{Y}_{{t}_{i-1}}\right]n{\left(\frac{\mathrm{log}n}{n}\right)}^{-\frac{1}{\alpha }}{h}^{2}\left(\stackrel{^}{\theta }-\theta \right)⇒\frac{v}{2\theta }\frac{{Z}_{1}}{{Z}_{0}}.$

$\frac{1}{n}\underset{i=1}{\overset{n-1}{\sum }}{Y}_{{t}_{i-1}}n{\left(\frac{\mathrm{log}n}{n}\right)}^{-\frac{1}{\alpha }}{h}^{2}\left(\stackrel{^}{\theta }-\theta \right)=\frac{1}{n}\underset{i=1}{\overset{n-1}{\sum }}{Y}_{{t}_{i-1}}\left[-\frac{{\Phi }_{1}\left(n\right)}{{\Phi }_{3}\left(n\right)}+\frac{{\Phi }_{2}\left(n\right)}{{\Phi }_{3}\left(n\right)}\right],$ (4.24)

5. 总结

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https://doi.org/10.1007/BF01460980

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https://doi.org/10.1007/s004400050082

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https://doi.org/10.1007/s004400100161

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[8] Benaïm, M. and Raimond, O. (2005) Self-Interacting Diffusions. III. Symmetric Interactions. Annals of Probability, 33, 1716-1759.
https://doi.org/10.1214/009117905000000251

[9] Benaïm, M. and Raimond, O. (2011) Self-Interacting Diffusions IV: Rate of Convergence. Electronic Journal of Probability, 16, 1815-1843.
https://doi.org/10.1214/EJP.v16-948

[10] Cranston, M. and Mountford, T.S. (1996) The Strong Law of Large Numbers for a Brownian Polymer. Annals of Probability, 24, 1300-1323.
https://doi.org/10.1214/aop/1065725183

[11] Chambeu, S. and Kurtzmann, A. (2011) Some Particular Self-Interacting Diffusions: Ergodic Behaviour and Almost Convergence. Bernoulli, 17, 1248-1267.
https://doi.org/10.3150/10-BEJ310

[12] Gauthier, C.-E. (2016) Self Attracting Diffusions on a Sphere and Application to a Periodic Case. Electronic Communications in Probability, 21, 1-12.
https://doi.org/10.1214/16-ECP4547

[13] Herrmann, S. and Roynette, B. (2003) Boundedness and Convergence of Some Self-Attracting Diffusions. Mathematische Annalen, 325, 81-96.
https://doi.org/10.1007/s00208-002-0370-0

[14] Herrmann, S. and Scheutzow, M. (2004) Rate of Convergence of Some Self-Attracting Diffusions. Stochastic Processes and Their Applications, 111, 41-55.
https://doi.org/10.1016/j.spa.2003.10.012

[15] Mountford, T. and Tarrés, P. (2008) An Asymptotic Result for Brownian Polymers. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 44, 29-46.
https://doi.org/10.1214/07-AIHP113

[16] Kleptsyny, V. and Kurtzmann, A. (2012) Ergodicity of Self-Attracting Motion. Electronic Journal of Probability, 17, 1-37.
https://doi.org/10.1214/EJP.v17-2121

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