A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile
| dc.creator | Hachem, Walid | |
| dc.creator | Loubaton, Philippe | |
| dc.creator | Najim, Jamal | |
| dc.date | 2007-06-01 | |
| dc.date.accessioned | 2026-07-07T08:03:53Z | |
| dc.date.available | 2026-07-07T08:03:53Z | |
| dc.description | Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $$ Y_{ij}^{n}=\frac{σ_{ij}(n)}{\sqrt{n}} X_{ij}^{n} $$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(σ_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$ \log\det(Y_n Y_n^* + ρI_N) $$ where $Y^*$ is the Hermitian adjoint of $Y$ and $ρ> 0$ is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4$^\textrm{th}$ moment of the $X_{ij}$'s differs from the 4$^{\textrm{th}}$ moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications. | |
| dc.identifier | https://arxiv.org/abs/0706.0166 | |
| dc.identifier | http://arxiv.org/abs/0706.0166 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/129747 | |
| dc.subject | Probability | |
| dc.subject | Primary 15A52, Secondary 15A18, 60F15 | |
| dc.title | A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile | |
| dc.type | text |