In this section, some properties of cumulants that will be used in the subsequent chapters are reviewed. For comprehensive treatment of higher-order statistics of random variables and stationary and nonstationary signals see (Brillinger 1965, 1981), (Rosenblatt 1974), (Gardner and Spooner 1994), (Spooner and Gardner 1994), (Dandawaté and Giannakis 1994, 1995), (Boashash et al. 1995), (Napolitano 1995), (Izzo and Napolitano 1998a), (Napolitano and Tesauro 2011).
Theorem 1.4.1 If two real-valued random-variable vectors and are statistically independent, then
Proof: Let , and . The characteristic function of the random vector X factorizes into the product of the characteristic functions of X1 and X2. That ...