A general correlation inequality for level sets of sums of independent random variables using the Bernoulli part with applications to the almost sure local limit theorem
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| Format: | Preprint |
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2023
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| _version_ | 1866917125494407168 |
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| author | Weber, Michel J. G. |
| author_facet | Weber, Michel J. G. |
| contents | Let $X=\{X_j , j\ge 1\}$ be a sequence of independent, square integrable variables taking values in a common lattice $\mathcal L(v_{ 0},D )= \{v_{ k}=v_{ 0}+D k , k\in \Z\}$.
Let $S_n=X_1+\ldots +X_n$, $a_n= {\mathbb E\,} S_n$, and $\s_n^2={\rm Var}(S_n)\to \infty$ with $n$. Assume that for each $j$, $\t_{X_j} =\sum_{k\in \Z}{\mathbb P}\{X_j=v_k\}\wedge{\mathbb P}\{X_j=v_{k+1}\}>0$. Using the Bernoulli part, we prove a general sharp correlation inequality extending the one we obtained in the i.i.d.\,case in \cite{W3}: Let $0<\t_j\le \t_{X_j}$ and assume that $ ν_n =\sum_{j=1}^n \t_j \, \uparrow \infty$, $n\to \infty$.
Let $\k_j\in \mathcal L(jv_0,D)$, $j=1,2,\ldots$ be a sequence of integers such that
\begin{equation*} {\rm(1)}\qquad\frac{κ_j-a_j}{\s_j}=\mathcal O(1 ), \qq\quad {\rm(2)}\qquad \s_j \,{\mathbb P}\{S_j=κ_j\} ={\mathcal O}(1).
\end{equation*}
Then there exists a constant $C $
such that for all $1\le m<n$, \begin{align*}
\s_n&\s_m \, \Big|{\mathbb P}\{S_n=\k_n, S_m=\k_m\}- {\mathbb P}\{S_n=\k_n \}{\mathbb P}\{ S_m=\k_m\} \Big|
\cr & \,\le \, \frac{C}{D^2}\, \max \Big(\frac{\s_n }{\sqrt{ν_n}},\frac{\s_m }{\sqrt {ν_m}} \Big)^3 \,\bigg\{ ν_n^{1/2} \prod_{j=m+1}^n\vartheta_j + {ν_n^{1/2} \over
(ν_n-ν_m) ^{3/2}}+{ 1\over \sqrt{ν_n\over ν_m}-1} \bigg\}.
\end{align*} We derive a sharp almost sure local limit theorem |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19372 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A general correlation inequality for level sets of sums of independent random variables using the Bernoulli part with applications to the almost sure local limit theorem Weber, Michel J. G. Probability 60F15, 60G50, 60F05 Let $X=\{X_j , j\ge 1\}$ be a sequence of independent, square integrable variables taking values in a common lattice $\mathcal L(v_{ 0},D )= \{v_{ k}=v_{ 0}+D k , k\in \Z\}$. Let $S_n=X_1+\ldots +X_n$, $a_n= {\mathbb E\,} S_n$, and $\s_n^2={\rm Var}(S_n)\to \infty$ with $n$. Assume that for each $j$, $\t_{X_j} =\sum_{k\in \Z}{\mathbb P}\{X_j=v_k\}\wedge{\mathbb P}\{X_j=v_{k+1}\}>0$. Using the Bernoulli part, we prove a general sharp correlation inequality extending the one we obtained in the i.i.d.\,case in \cite{W3}: Let $0<\t_j\le \t_{X_j}$ and assume that $ ν_n =\sum_{j=1}^n \t_j \, \uparrow \infty$, $n\to \infty$. Let $\k_j\in \mathcal L(jv_0,D)$, $j=1,2,\ldots$ be a sequence of integers such that \begin{equation*} {\rm(1)}\qquad\frac{κ_j-a_j}{\s_j}=\mathcal O(1 ), \qq\quad {\rm(2)}\qquad \s_j \,{\mathbb P}\{S_j=κ_j\} ={\mathcal O}(1). \end{equation*} Then there exists a constant $C $ such that for all $1\le m<n$, \begin{align*} \s_n&\s_m \, \Big|{\mathbb P}\{S_n=\k_n, S_m=\k_m\}- {\mathbb P}\{S_n=\k_n \}{\mathbb P}\{ S_m=\k_m\} \Big| \cr & \,\le \, \frac{C}{D^2}\, \max \Big(\frac{\s_n }{\sqrt{ν_n}},\frac{\s_m }{\sqrt {ν_m}} \Big)^3 \,\bigg\{ ν_n^{1/2} \prod_{j=m+1}^n\vartheta_j + {ν_n^{1/2} \over (ν_n-ν_m) ^{3/2}}+{ 1\over \sqrt{ν_n\over ν_m}-1} \bigg\}. \end{align*} We derive a sharp almost sure local limit theorem |
| title | A general correlation inequality for level sets of sums of independent random variables using the Bernoulli part with applications to the almost sure local limit theorem |
| topic | Probability 60F15, 60G50, 60F05 |
| url | https://arxiv.org/abs/2305.19372 |