Functional Equations and Characterization Problems on Locally Compact Abelian Groups
Characterization theorems in mathematical statistics started with the celebrated Kac-Bernstein characterization result, which dates back to 1939 and 1949, stating that the independence of the sum χ1 + χ2 and of the difference χ1 - χ2 of independent identically distributed L2-random variables χ1 and χ2 implies that the variables χj’ are Gaussian. The statement was the first among those saying (under some restrictions) that independent linear forms of independent random variables χj may exist only for Gaussian χj’. Skitovich and Darmois (1953) solved the problem completely in dimension one and the multidimensional extension was provided by Ghurye and Olkin in 1962. The monograph Characterization theorems in mathematical statistics by A. M. Kagan, Yu. V. Linnik and C. R. Rao (J. Wiley, 1973) provides a full account of achievements in Rd settings. In recent years, the above topic has been extensively studied for random variables with values in more general algebraic structures (such as locally compact Abelian groups, Lie and quantum groups and symmetric spaces). This book contains generalizations of the Kac-Bernstein and Skitovich-Darmois theorems to locally compact Abelian groups and, in particular, the characterization of Gaussian and idempotent probability distributions. Solutions to these problems are transformed to solutions of some functional equations in the set of positive definite functions on the corresponding character group. The self-contained and well-written monograph is aimed at mathematicians interested in probability on algebraic structures and abstract harmonic analysis. Future research may be stimulated by a collection of unsolved problems at the end of the book.