论文标题
$ m $依赖性随机变量的中心极限定理中的定量界限
Quantitative bounds in the central limit theorem for $m$-dependent random variables
论文作者
论文摘要
对于每个$ n \ ge 1 $，令$ x_ {n，1}，\ ldots，x_ {n，n_n} $为真实的随机变量，$ s_n = \ sum_ {i = 1}^{n_n} {n_n} x_ {x_ {x_ {n，i} $。令$ m_n \ ge 1 $为整数。假设$（x_ {n，1}，\ ldots，x_ {n，n_n}）$是$ m_n $ - depentent，$ e（x_ {ni}）= 0 $，$ e（x_ {x_ {ni}^2）<\ infty $ and $ ni $ = e（ni}^2）然后，\ b begin {chater*} d_w \ bigl（\ frac {s_n} {σ_n}，\，\，z \ bigr）\ le 30 \，\ bigl \ {c^{1/3} {1/3} } n \ ge 1 \ text {and} c> 0，\ end {chater*}其中$ d_w $是wasserstein decorts，$ z $ a标准正常随机变量和$$ u_n（c）= \ frac {m_n} {σ_n^2} \，\ sum_ {i = 1}^{n_n} e \ bigl [x_ {x_ {n ，i}^2 \，1 \ bigl \ {\ abs {x_ {n，i}}> c \，σ_n/m_n \ bigr \} \ bigr]。$$除其他事项外，$ d_w \ bigl（s_n/σ_n，\，z \ bigr）$的估计值产生的$ d_ {tv} \ bigl的类似估计值（s_n/σ_n，\，z \ bigr）在$ d_ {tv tv tev} $的情况下是总变体距离。
For each $n\ge 1$, let $X_{n,1},\ldots,X_{n,N_n}$ be real random variables and $S_n=\sum_{i=1}^{N_n}X_{n,i}$. Let $m_n\ge 1$ be an integer. Suppose $(X_{n,1},\ldots,X_{n,N_n})$ is $m_n$-dependent, $E(X_{ni})=0$, $E(X_{ni}^2)<\infty$ and $σ_n^2:=E(S_n^2)>0$ for all $n$ and $i$. Then, \begin{gather*} d_W\Bigl(\frac{S_n}{σ_n},\,Z\Bigr)\le 30\,\bigl\{c^{1/3}+12\,U_n(c/2)^{1/2}\bigr\}\quad\quad\text{for all }n\ge 1\text{ and }c>0, \end{gather*} where $d_W$ is Wasserstein distance, $Z$ a standard normal random variable and $$U_n(c)=\frac{m_n}{σ_n^2}\,\sum_{i=1}^{N_n}E\Bigl[X_{n,i}^2\,1\bigl\{\abs{X_{n,i}}>c\,σ_n/m_n\bigr\}\Bigr].$$ Among other things, this estimate of $d_W\bigl(S_n/σ_n,\,Z\bigr)$ yields a similar estimate of $d_{TV}\bigl(S_n/σ_n,\,Z\bigr)$ where $d_{TV}$ is total variation distance.
