Approximation theorems for random permanents and associated stochastic processes
Let \({\mathbf X}^{(n)}= [X_{ij}]\) be an \(m\times n\) \((m\leq n)\) real random matrix of square integrable components and such that its columns are built from the first terms of i.i.d. sequences \((X_{ij})_{i\geq 1}\) \((1\leq j\leq n)\) of exchangeable random variables. Define the permanent of \({\mathbf X}^{(n)}\) by \[ \text{Per\,}{\mathbf X}^{(n)}= \sum_{(i_1,\dots, i_m)\subset\{1,\dots, n\}} X_{1i_1}\cdots X_{mi_m}. \] Let \(\mu= EX_{ij}\neq 0\), \(\sigma^2= \text{Var\,}X_{i,j}\) and \(\rho= \text{corr}(X_{kj}, X_{ij})\). Define the stochastic process associated with \(\text{Per\,}{\mathbf X}^{(n)}\) by the relation \[ {\mathcal P}{\mathcal S}{\mathcal P}{\mathbf X}^{(n)}(t)= {n\choose m} m!\mu^m\left(1+ \sum^m_{c=1} {m\choose c} U^{(m,n)}_c(t)\right)\quad (t\in [0,1]), \] where \[ U^{(m,n)}_c(t)= {1\over{n\choose c}{m\choose c}}{1\over c!} \sum_{1\leq i_1<\cdots< i_c\leq m}\,\sum_{1\leq j_1<\cdots< j_c\leq [nt]} \text{Per}(\widetilde X_{i_u j_v})_{_{\substack{ u= 1,\dots, c\\ v=1,\dots,c}}} \] for \(\widetilde X_{ij}= X_{ij}/\mu- 1\) (\(1\leq i\leq m\), \(1\leq j\leq n\)). Establishing the limiting result for \(U^{(m,n)}_c(\cdot)\) \((c\geq 1)\) the authors obtain functional limit theorems for the process \(({\mathcal P}{\mathcal S}{\mathcal P}{\mathbf X}^{(n)}(t))_{t\in [0,1]}\) such as \[ ({\mathcal P}{\mathcal S}{\mathcal P}{\mathbf X}^{(n)}(t))_{t\in [0,1]}@>d>>\Biggl(\exp\Biggl(\sqrt{\lambda}\gamma B_t- {\lambda t\gamma^2\over 2}\Biggr)\Biggr)_{t\in [0,1]}\quad\text{in }D_{\mathbf R} \] if \(\rho= 0\) and \(m/n\to \lambda> 0\) as \(n\to\infty\), where \((B_t)_{t\in [0,1]}\) is the standard Brownian motion and \(\gamma= \sigma/\mu\).
random matrices martingale decomposition
scientific article; zbMATH DE number 2176189
scientific article