AI 中文总结
本文首次将Ghurye--Olkin--Ibragimov定理推广到第二可数局部紧阿贝尔群及$\mathbf{a}$-adic螺线管上，研究独立随机变量序列线性形式的独立性刻画，系数为拓扑自同构。
AI 中文摘要
根据经典的Skitovich--Darmois定理，实直线上的高斯分布由有限个独立随机变量$\xi_j$的两个线性形式的独立性所刻画。该定理已在多个方向上得到推广。特别地，S.G. Ghurye和I. Olkin建立了当$\xi_j$为$n$维独立随机向量且线性形式的系数为可逆的$n\times n$矩阵时的类似结果。随后，A.A. Zinger以及后来的I.A. Ibragimov研究了$n$维独立随机向量的无穷序列的线性形式。在本文中，我们首次研究取值于第二可数局部紧阿贝尔群$X$的独立随机变量序列的线性形式，在以下任一条件下：$X$不包含非平凡的紧子群，或$X$不包含拓扑同构于圆周群的子群且具有有限的拓扑自同构群。此外，我们研究了独立随机变量取值于$\mathbf{a}$-adic螺线管$\Sigma_{\mathbf{a}}$的情形。在所有上述设定中，线性形式的系数均为相应群的拓扑自同构。
英文摘要
According to the classical Skitovich--Darmois theorem, the Gaussian distribution on the real line is characterized by the independence of two linear forms of a finite number of independent random variables $ξ_j$. This theorem has been extended in various directions. In particular, S.G. Ghurye and I. Olkin established an analogous result for the case where $ξ_j$ are $n$-dimensional independent random vectors and the coefficients of the linear forms are invertible $n\times n$ matrices. Subsequently, A.A. Zinger and later I.A. Ibragimov investigated linear forms of an infinite sequence of $n$-dimensional independent random vectors. In the present paper, we investigate, for the first time, linear forms of a sequence of independent random variables taking values in a second-countable locally compact Abelian group $X$ under either of the following conditions: $X$ contains no nontrivial compact subgroups, or $X$ contains no subgroups topologically isomorphic to the circle group and has a finite topological automorphism group. Furthermore, we investigate the case where the independent random variables take values in an $\mathbf{a}$-adic solenoid $Σ_{\mathbf{a}}$. In all these settings, the coefficients of the linear forms are topological automorphisms of the corresponding group.
Comments15 pages
