The Marcinkiewicz-Zygmund law of large numbers on the group of Euclidean motions and the diamond group
Let \(H^d=R^d\times R^d\times R\) be the \((2d+1)\)-dimensional Heisenberg group, let \(\text{Sp} (d)\) be the symplectic group of \(R^{2d}\times\{0\}\subset H^d\), and let \(\text{O} (2d)\) be the orthogonal group of \(R^{2d}\times\{0\}\subset H^d\). Then the diamond motion group \(\text{DM} _d\) is defined as the semidirect product \(H^d\otimes (\text{Sp} (d)\cap \text{O} (2d))\), equipped with the operation \((x,u)\cdot (y,v):=(x\cdot u(y),u\cdot v)\), \(x,y\in H^d\), \(u, v\in \text{Sp} (d)\cap \text{O} (2d)\). Let \(\pi (x)\) be the projection of \(x\in \text{DM} _d\) onto \(H^d\). For \(a>0\), define the automorphism \(\delta_a\) of \(\text{DM} _d\) by \(\delta_a(x,u):=(ax',ax'',a^2x''',u)\) for \(x=(x',x'',x''')\in R^d\times R^d\times R\), \(u\in \text{Sp} (d)\cap \text{O} (2d)\). Let \(\{X_n:n\geq 1\}=\{(Y_n,U_n):n\geq 1\}= \{(Y_n',Y_n'',Y_n''',U_n):n\geq 1\}\) be independent identically distributed random variables with values in the diamond motion group \(\text{DM} _d\). It is proved that, for any \(0<p<2\), \(\pi (\delta_{n^{-1/p}}(\prod_{i=1}^n(c\cdot X_i))\to 0\) almost surely as \(n\to\infty\) for some \(c=(y,u)=(y',y'',y''',u) \in \text{DM} _d\) if and only if \(E| Y_1| ^p<\infty\). In the case \(1\leq p<2\), \(c=(-u(EY_1',EY_1''),y''',u)\) for arbitrary \(y'''\in R\) and \(u\in \text{Sp} (d)\cap \text{O} (2d)\). In the case \(0<p<1\), \(c\) may be chosen arbitrarily.
scientific article; zbMATH DE number 1213224