论文标题
在Rozanov的定理上，并进行渐近均匀分布
On Rozanov's Theorem and strenghtened asymptotic uniform distribution
论文作者
论文摘要
对于sums $ s_n = \ sum_ {k = 1}^n x_k $，$ n \ ge 1 $的独立随机变量的$ x_k $ in $ \ z $ 由于更一般的结果，我们证明，如果（i）对于某些功能$ 1 \ lex.luparrow \ uparrow \ infty $ as $ t \ to \ infty $，而某些常数$ c $，我们都有$ n $ and $ n $和$ν\ in \ z $，in \ z $， \ begin {equination*} \ label {Abstract1} \ big |b_n¶\ big \ {s_n = v务{c \ over \，ϕ（b_n）}， \ end {equation*} 然后（ii）存在数值常数$ C_1 $，因此对于所有$ n $ 和$ \ m = 0,1，\ ldots，h-1 $，\ begin {align*} \ label {Abstract1} \ big | {\ mathbb p} \ big \ {s_n \ equiv \，\ m \ \ \ hbox {\ rm {（mod $ h $）}}} \ big \} - \ frac {1} {1} {h} {h} {h} \ le {1 \ over \ sqrt {2π} \，b_n}+ \ frac {1+ 2 {c}/{h}} {ϕ（b_n）^{2/3}}}}}}}}+ c_1+ c_1+ c_1+ c_1 \，e^{ - （1/16）ϕ（1/16）ϕ（b_n）（b_n） \ end {align*}假设（i）是否适用于通常形式的局部限制定理，并且（ii）产生了Rozanov必要条件的影响。 假设代替（i），$ \ t_j = \ sum_ {k \ in \ z} {\ mathbb p} \ {x_j = k \} \ wedge {\ mathbb p} \ {x_j = k = k+1 \} $ν_n= \ sum_ {j = 1}^n \ t_j \ uparrow \ infty $。 我们还证明了渐近均匀分布特性的形式。
For sums $S_n=\sum_{k=1}^n X_k$, $n\ge 1$ of independent random variables $ X_k $ taking values in $\Z$ we prove, as a consequence of a more general result, that if (i) For some function $1\le ϕ(t)\uparrow \infty $ as $t\to \infty$, and some constant $C$, we have for all $n$ and $ν\in \Z$, \begin{equation*}\label{abstract1} \big|B_n¶\big\{ S_n=ν\big\}- {1\over \sqrt{ 2π} }\ e^{- {(ν-M_n)^2\over 2 B_n^2} }\big|\,\le \, {C\over \,ϕ(B_n)}, \end{equation*} then (ii) There exists a numerical constant $C_1$, such that for all $n $ such that $B_n\ge 6$, all $h\ge 2$, and $\m=0,1,\ldots, h-1$, \begin{align*}\label{abstract1} \Big|{\mathbb P}\big\{ S_n\equiv\, \m\ \hbox{\rm{ (mod $h$)}}\big\}- \frac{1}{h}\Big| \le {1\over \sqrt{2π}\, B_n }+\frac{1+ 2 {C}/{h} }{ ϕ(B_n)^{2/3} } + C_1 \,e^{-(1/ 16 )ϕ(B_n)^{2/3}}. \end{align*} Assumption (i) holds if a local limit theorem in the usual form is applicable, and (ii) yields a strenghtening of Rozanov's necessary condition. Assume in place of (i) that $\t_j =\sum_{k\in \Z}{\mathbb P}\{X_j= k\}\wedge{\mathbb P}\{X_j= k+1 \} >0$, for each $j$ and that $ν_n =\sum_{j=1}^n \t_j\uparrow \infty$. We prove also strenghtened forms of the asymptotic uniform distribution property.
