On the best constants in Khintchine type inequalities for martingales
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918255749234688 |
|---|---|
| author | Karagulyan, Grigori A. |
| author_facet | Karagulyan, Grigori A. |
| contents | For discrete martingale-difference sequences $d=\{d_1,\ldots,d_n\}$ we consider Khintchine type inequalities, involving certain square function $\mathfrak S (d)$ considered by Chang-Wilson-Wolff in 1982. In particular, we prove
\begin{equation}
\left\|\sum_{k=1}^nd_k\right\|_p\le 2^{1/2}\big(Γ((p+1)/2))/\sqrtπ\big)^{1/p}\|\mathfrak S(d)\|_\infty,\quad p\ge 3,
\end{equation}
where the constant on the right hand side is the best possible and the same as known for the Rademacher sums $\sum_{k=1}^na_kr_k$. Moreover, for a fixed $n$ the constant in the inequality can be replaced by $\sum_{k=1}^nr_k/\sqrt{n}$. We apply a technique, reducing the general case to the case of Haar and Rademacher sums, that allows also establish a sub-Gaussian estimate
\begin{equation}
{\bf E}\left[\exp\left(λ\cdot \left(\frac{\sum_{k=1}^nd_k}{\|\mathfrak S(d)\|_\infty}\right)^2\right)\right]\le \frac{1}{\sqrt{1-2λ}},\quad 0<λ<1/2,
\end{equation}
where the constant on the right hand side is the best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the best constants in Khintchine type inequalities for martingales Karagulyan, Grigori A. Probability Classical Analysis and ODEs 42C05, 42C10, 60G42 For discrete martingale-difference sequences $d=\{d_1,\ldots,d_n\}$ we consider Khintchine type inequalities, involving certain square function $\mathfrak S (d)$ considered by Chang-Wilson-Wolff in 1982. In particular, we prove \begin{equation} \left\|\sum_{k=1}^nd_k\right\|_p\le 2^{1/2}\big(Γ((p+1)/2))/\sqrtπ\big)^{1/p}\|\mathfrak S(d)\|_\infty,\quad p\ge 3, \end{equation} where the constant on the right hand side is the best possible and the same as known for the Rademacher sums $\sum_{k=1}^na_kr_k$. Moreover, for a fixed $n$ the constant in the inequality can be replaced by $\sum_{k=1}^nr_k/\sqrt{n}$. We apply a technique, reducing the general case to the case of Haar and Rademacher sums, that allows also establish a sub-Gaussian estimate \begin{equation} {\bf E}\left[\exp\left(λ\cdot \left(\frac{\sum_{k=1}^nd_k}{\|\mathfrak S(d)\|_\infty}\right)^2\right)\right]\le \frac{1}{\sqrt{1-2λ}},\quad 0<λ<1/2, \end{equation} where the constant on the right hand side is the best possible. |
| title | On the best constants in Khintchine type inequalities for martingales |
| topic | Probability Classical Analysis and ODEs 42C05, 42C10, 60G42 |
| url | https://arxiv.org/abs/2401.16153 |