Limit theorems for mixed-norm sequence spaces with applications to volume distribution
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909380932272128 |
|---|---|
| author | Juhos, Michael Kabluchko, Zakhar Prochno, Joscha |
| author_facet | Juhos, Michael Kabluchko, Zakhar Prochno, Joscha |
| contents | Let $p, q \in (0, \infty]$ and $\ell_p^m(\ell_q^n)$ be the mixed-norm sequence space of real matrices $x = (x_{i, j})_{i \leq m, j \leq n}$ endowed with the (quasi-)norm $\Vert x \Vert_{p, q} := \big\Vert \big( \Vert (x_{i, j})_{j \leq n} \Vert_q \big)_{i \leq m} \Vert_p$. We shall prove a Poincaré-Maxwell-Borel lemma for suitably scaled matrices chosen uniformly at random in the $\ell_p^m(\ell_q^n)$ unit balls $\mathbb{B}_{p, q}^{m, n}$, and obtain both central and non-central limit theorems for their $\ell_p(\ell_q)$-norms. We use those limit theorems to study the asymptotic volume distribution in the intersection of two mixed-norm sequence balls. Our approach is based on a new probabilistic representation of the uniform distribution on $\mathbb{B}_{p, q}^{m, n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_08937 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Limit theorems for mixed-norm sequence spaces with applications to volume distribution Juhos, Michael Kabluchko, Zakhar Prochno, Joscha Probability Functional Analysis 52A23, 60F05 (Primary) 46B06, 60D05 (Secondary) Let $p, q \in (0, \infty]$ and $\ell_p^m(\ell_q^n)$ be the mixed-norm sequence space of real matrices $x = (x_{i, j})_{i \leq m, j \leq n}$ endowed with the (quasi-)norm $\Vert x \Vert_{p, q} := \big\Vert \big( \Vert (x_{i, j})_{j \leq n} \Vert_q \big)_{i \leq m} \Vert_p$. We shall prove a Poincaré-Maxwell-Borel lemma for suitably scaled matrices chosen uniformly at random in the $\ell_p^m(\ell_q^n)$ unit balls $\mathbb{B}_{p, q}^{m, n}$, and obtain both central and non-central limit theorems for their $\ell_p(\ell_q)$-norms. We use those limit theorems to study the asymptotic volume distribution in the intersection of two mixed-norm sequence balls. Our approach is based on a new probabilistic representation of the uniform distribution on $\mathbb{B}_{p, q}^{m, n}$. |
| title | Limit theorems for mixed-norm sequence spaces with applications to volume distribution |
| topic | Probability Functional Analysis 52A23, 60F05 (Primary) 46B06, 60D05 (Secondary) |
| url | https://arxiv.org/abs/2209.08937 |