Functional central limit theorems on Lie groups. A survey (Q2769700)

The functional central limit problem on a Lie group \(G\) can be formulated as follows. There is given a rowwise independant array \(\{\xi_{nl}: (n,l)\in{\mathbb N}^2\}\) of \(G\)-valued random variables which for a sequence \(\{k_n: n\in{\mathbb N}\}\) of increasing right-continuous scaling functions \(k_n: {\mathbb R}_+\to{\mathbb Z}_+\) with \(k_n(0)=0\) satisfies the infinitesimality condition NEWLINE\[NEWLINE \lim_{n\to\infty}\max_{1\leq l\leq k_n(t)} {\mathbb P}(\xi_{nl}\notin U) = 0 NEWLINE\]NEWLINE for all Borel neighbourhoods \(U\) of the neutral element of \(G\) and for all \(t\in{\mathbb R}_+\). One forms the products \(\xi_n(t)=\prod_{l=1}^{k_n(t)}\xi_{nl}\) and considers the sequence \(\{\xi_n:\;n\in{\mathbb N}\}\) of stochastic processes with paths in the Skorokhod space \({\mathbb D}({\mathbb R}_+,G)\) of càdlàg functions. One searches for conditions on the array and the scaling functions so that convergence \(\xi_n\to\xi\) in distribution in the Skorokhod space holds, where \(\xi=\{\xi(t):t\in{\mathbb R}_+\}\) is a process, necessarily having independent (but not necessarily) stationary left-increments. The general solution of the functional central limit problems for triangular arrays of random variables with values in a Lie group is described. The role of processes of finite variation is clarified. The special case of processes with independent increments having Markov generator is treated. Connections with Hille-Yosida theory for two-parameter evolution families of operators and with the martingale problem are explained.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].