论文标题
I.I.D的总和沿倍数组的一组均匀的半本地限制定理。随机变量
A uniform semi-local limit theorem along sets of multiples for sums of i.i.d. random variables
论文作者
论文摘要
令$ x $为一个正方形的随机变量，带有基本概率空间$（Ø，\ a，¶）$，在晶格$ \ Mathcal l（v_0,1）= \ big \ \ big \ {v_0+ k，k \ in \ z \ k \ big \} $和SUM__x = \ sum___ { \ z}¶\ {x = v_k \} \wedge¶\ {x = v_ {k+1} \}> 0 $。令$ x_i $，$ i \ ge 1 $是独立的，与$ x $相同的法律分布相同的随机变量，让$ s_n = \ sum_ {j = 1}^nx_j $，对于每个$ n $。令$ \ m_k \ ge 0 $为$ \ m = \ sum_ {k \ in \ z} \ m_k $验证$ 1- \ t_x <\ m <1 $，注意$ \ t_x <1 $始终。再说$ \ t = 1- \ m $，$ s（t）= \ sum_ {k \ in \ z} \ m_k \，e^{2iπv_kt} $，$ρ$使得$ 1- \ t <ρ<ρ<1 $。 我们证明了类$ \ mathcal f = \ {f_ {d}，d \ ge 2 \} $的以下统一的半本地定理，其中$ f_ {d} = d \ n $。 \ noi（i）存在$θ=θ（ρ，\ t）$，$ 0 <θ<\ t $，$ c $和$ n $ 因此，对于$ n \ ge n $，\ begin {align*} \ sup_ {u \ ge 0} \，\ sup_ {d \ ge 2} \ big | ¶\ {s_n+u \在f_ {d} \}中 - {1 \ over d} - {1 \ over d} \ sum_ {0 <| \ el | <d} \ e \，e^{2iπ{\ ell \ d} \ widetilde x} +s \ big（{\ ell \ vos d} \ big）\ big）^n \ big | \ cr＆\ le \ frac {c} {θ^{3/2}}} \ \ \\ frac {（\ log n）^{5/2}}} {n^{3/2}}}}+2ρ^n。 \ end {align*} \ vskip 1 pt \ noi（ii）让$ \ mathcal d $是除数的测试集$ \ ge 2 $，$ \ MATHCAL d_ \ p $是$ \ MATHCAL D $的部分高度$ \ p $ \ p $和$ | \ \ | \ MATHCAL D_ \ p | $ d_ \ p | $ neote its ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS ITS。然后， \ begin {eqnarray*} \ sum_ {n = n}^\ infty \ \ sup_ { ¶\ {d | s_n+u \} - {1 \ fover d} \ big | ＆\ le＆\ frac {c_1} {\ t} \，， + \ frac {c_2} {θ^{3/2}}} + \ frac {2ρ^2} {1-ρ}。 \ end {eqnarray*}
Let $X $ be a square integrable random variable with basic probability space $(Ø, \A, ¶)$, taking values in a lattice $\mathcal L(v_0,1)=\big\{v_k=v_0+ k,k\in \Z\big\}$ and such that $\t_X =\sum_{k\in \Z}¶\{X=v_k\}\wedge ¶\{X=v_{k+1}\}>0$. Let $ X_i$, $i\ge 1 $ be independent, identically distributed random variables having same law than $X$, and let $S_n=\sum_{j=1}^nX_j$, for each $n$. Let $\m_k\ge 0$ be such that $ \m= \sum_{k\in \Z}\m_k $ verifies $1- \t_X<\m<1$, noting that $\t_X< 1$ always. Further let $\t=1-\m$, $s(t) =\sum_{k\in \Z} \m_k\, e^{ 2i πv_kt}$ and $ρ$ be such that $1-\t<ρ<1$. We prove the following uniform semi-local theorems for the class $\mathcal F=\{F_{d}, d\ge 2\}$, where $F_{d}= d\N$. \noi(i) There exists $θ=θ(ρ,\t)$ with $ 0< θ<\t$, $C$ and $N$ such that for $ n \ge N$, \begin{align*} \sup_{u\ge 0}\,\sup_{d\ge 2} \Big| ¶\{ S_n+u\in F_{d} \} - {1\over d}- {1\over d}\sum_{ 0< |\ell|<d }& \Big( e^{ (iπ{\ell\over d }-{ π^2\ell^2\over 2 d^2}) } \t\, \E \,e^{2i π{\ell\over d }\widetilde X } +s\big( {\ell\over d }\big)\Big)^n \Big| \cr &\le \frac{C }{ θ^{3/2}}\ \frac{(\log n)^{5/2}}{ n^{3/2}}+2ρ^n. \end{align*} \vskip 1 pt \noi(ii) Let $\mathcal D$ be a test set of divisors $\ge 2$, $\mathcal D_\p$ be the section of $\mathcal D$ at height $\p$ and $|\mathcal D_\p|$ denote its cardinality. Then, \begin{eqnarray*} \sum_{n=N}^\infty \ \sup_{u\ge 0} \, \sup_{\p\ge 2}\, {1\over |\mathcal D_\p |} \sum_{d\in \mathcal D_\p } \,\Big| ¶\{d|S_n+u \} - {1\over d}\Big| & \le & \frac{C_1}{\t} \, + \frac{C_2 }{ θ^{3/2}} +\frac{2ρ^2}{1-ρ}. \end{eqnarray*}
