Characterization theorems for \(Q\)-independent random variables with values in a Banach space (Q826689)

Consider a random variable \(\xi\) taking values in a real, separable Banach space \(X\). Such a random variable is said to be Gaussian if \(\langle\xi,f\rangle\) is Gaussian for all \(f\) in the dual space \(X^*\). A sequence \(\xi_1,\ldots,\xi_n\) of random variables taking values in \(X\) is said to be \(Q\)-independent if \[ \mathbb{E}[e^{i\langle(\xi_1,\ldots,\xi_n),(f_1,\ldots,f_n)\rangle}] = \left(\prod_{j=1}^n\mathbb{E}[e^{i\langle\xi_j,f_j\rangle}]\right)e^{q(f_1,\ldots,f_n)}\,, \] for all \(f_1,\ldots,f_n\in X^*\), where \(q(f_1,\ldots,f_n)\) is a continuous polynomial on \((X^*)^n\) with \(q(0,\ldots,0)=0\). In this paper, the author establishes a number of characterization theorems for Gaussianity of a sequence of \(Q\)-independent random variables taking values in \(X\). These include analogues of the Skitovich-Darmois theorem (characterizing Gaussianity based on \(Q\)-independence of linear forms) and the Heyde theorem (characterizing Gaussianity based on symmetry of conditional distributions). Finally, the author also proves that \(Q\)-independence of the sample mean \(S=\frac{1}{n}\sum_{i=1}^n\xi_i\) and the residual vector \((\xi_1-S,\ldots,\xi_n-S)\) is sufficient for Gaussianity of a sequence of \(Q\)-independent and identically distributed random variables.