Laguerre calculus and its applications to probability measures on \(H^n\) (Q2752164)

Let \(H^n\) denote the \((2n+1)\)-dimensional Heisenberg group represented as \(\mathbb{C}\otimes \mathbb{R}\) and endowed with the natural group of dilations \((\delta_t)_{t>0}\) and a homogeneous norm. The Fourier transform on the centre \((\approx\mathbb{R})\) defines the partial transform operator \(f\mapsto \widetilde f_\tau\) (for \(\tau\in \mathbb{R}^*)\) and twisted convolution \(*_\tau\) in the usual way. The Laguerre polynomials \((L_k^{(0)})\) define an ONS system \((W_n^{(0)})\) on \(L^2(\mathbb{C}^n)\). The Laguerre calculus is first used to obtain a representation of the solutions of the heat equations \({\partial u\over \partial s}+ {\mathcal L}_\alpha \equiv 0\) [where \({\mathcal L}_\alpha= -{1\over 2}\sum (Z_j\overline Z_j+ \overline Z_jZ_j) +i\alpha T\); see \textit{G. B. Folland} and \textit{E. M. Stein}, Comm. Pure Appl. Math. 27, 429-522 (1974; Zbl 0293.35012)]. NEWLINENEWLINENEWLINESection 3 is concerned with central limit theorems (CLT) on \(H^n\): By the above mentioned methods it is proved that \(\delta_{1/ \sqrt k} (\mu^k) \to\nu\) for centred probabilities fulfilling a Lindeberg condition. As corollaries, for non-centred \(\mu\), the authors prove CLT for shifted laws. It should be mentioned that there exist similar investigations of CLT on homogeneous groups (with different methods) [see e.g. \textit{G. Pap}, Probab. Math. Stat. 14, 287-312 (1993; Zbl 0829.60004) or \textit{H.-P. Scheffler}, J. Theor. Probab. 7, 767-792 (1994; Zbl 0807.60010)] and the survey on the literature mentioned there.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00044].