The Prokhorov-Loève law of large numbers on countable subgroups of simply connected step 2-nilpotent Lie groups (Q1288943)

A random variable \(X\) on a locally compact group is called symmetric if both \(X\) and \(X^{-1}\) have the same distribution \(\alpha(X)\). The main result of this note is Theorem 2: Let \(G\) be a countable subgroup of a \(d\)-dimensional simply connected step 2-nilpotent Lie group. Assume that \((X_n)_{n\geq 1}\) is a sequence of independent symmetric \(G\)-valued random variables. Let, furthermore, \(\{A_n\}_{n\geq 1}\) be a sequence of endomorphisms of \(G\). For \[ A_n \prod^n_{k=1} X_k @>\text{a.s.}>> 0,\quad n\to\infty, \] it is necessary and sufficient that both of the following two conditions hold: \[ A_n X_k @>\text{a.s.}>> 0,\quad n\to\infty,\quad k\geq 1\quad \text{and}\quad A_n \prod^{n_{j+ 1}}_{k= n_j+ 1} X_k @>\text{a.s.}>> 0,\quad j\to\infty, \] for any nondecreasing unbounded sequence \(\{n_j\}_{j\geq 1}\) of natural numbers. For details, see the author's references.