Tauberian theorems for matrix regular variation (Q2839392)

Let \(\mathrm{GL}(\mathbb R^{m})\) denote the space of invertible \(m\times m\) matrices with real entries. A Borel measurable function \(U:\mathbb R^{+}\rightarrow \mathrm{GL}(\mathbb R^{m}) \) is regularly varying with index matrix \(E\) if \(U(tx)U(x)^{-1}\rightarrow t^{E}\) as \(x\rightarrow \infty \); notation \(U(x)\in RV(E)\). Assume that the Laplace-Stieltjes transform \(\widehat{U}(s)=\int_{0}^{\infty }e^{-sx}dU(x)\) exists for \(s>0\). For functions of bounded variation, the authors prove the following theorem. Suppose that every eigenvalue of \(E\) has a positive real part. Then \(U(x)\in RV(E)\) if and only if \(\widehat{U}(1/x)\in RV(E)\) and in either case, \(\widehat{U}(1/x)U(x)^{-1}\rightarrow \Gamma (E+I)\), where \(I\) denotes the identity matrix and \(\Gamma (.)\) the matrix gamma function. The results could find applications in time series analysis.