【摘要】 八元数偏移线性正则变换(octonion offset linear canonical transform, OOLCT)作为八元数线性正则变换(octonion linear canonical transform, OLCT)的推广形式，在高维非平稳信号时频调控中具有优势，但八元数非结合性使传统概率统计量定义失效，且现有研究多聚焦四元数域，缺乏三维OOLCT(3D-OOLCT)域的严谨概率框架。本文将基础概率理论引入3D-OOLCT领域，构建兼容八元数特性的概率体系：首先，定义3D-OOLCT域中八元数值概率密度函数、分布函数、均值及特征函数；其次，证明特征函数定理；最后，通过算例推导验证特定概率密度函数在3D-OOLCT域的特征函数表达式。该研究填补OOLCT域与概率理论结合的空白，完善八元数变换理论，为三维高维随机信号统计分析提供新工具，也为后续相关工程应用奠定数理基础。更多还原

【Abstract】 Octonion offset linear canonical transform(OOLCT), as a generalized form of octonion linear canonical fransform(OLCT), has advantages in time-frequency manipulation of high dimensional signals. However, non-associative nature of octonions renders traditional definitions of probability statistics invalid. Moreover, existing studies mostly focus on the quaternion domain and lack a rigorous probabilistic framework in the three-dimensional OOLCT(3D-OOLCT) domain. This paper introduces basic probability theory into the field of 3D-OOLCT and constructs a probability system compatible with the characteristics of octonions. First, the octonion numerical probability density function, distribution function, mean, and characteristic function in the 3D-OOLCT domain are defined. Second, the characteristic function theorem is proved. Finally, the characteristic function expression of a specific probability density function in the 3D-OOLCT domain is derived and verified through numerical examples. This research fills a gap in the integration of OOLCT domains with probability theory, improves the theory of octonion transformations, provides new tools for the statistical analysis of three-dimensional and high-dimensional random signals, and lays a mathematical foundation for subsequent related engineering applications.更多还原