On Rozanov's Theorem and strenghtened asymptotic uniform distribution
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913419580407808 |
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| author | Weber, Michel J. G. |
| author_facet | Weber, Michel J. G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12228 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Rozanov's Theorem and strenghtened asymptotic uniform distribution Weber, Michel J. G. Probability Primary: 60F15, 60G50, Secondary: 60F05 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. |
| title | On Rozanov's Theorem and strenghtened asymptotic uniform distribution |
| topic | Probability Primary: 60F15, 60G50, Secondary: 60F05 |
| url | https://arxiv.org/abs/2209.12228 |