defined as: And the pseudo-cross-covariance matrix is defined as: Two complex random vectors and are called uncorrelated if . Independence Two complex random vectors and are called independent if where and denote the cumulative distribution functions of and as defined in and denotes their joint cumulative distribution function. Independence of and is often denoted by . Written component-wise, and are called independent if . Circular symmetry A complex random vector is called circularly symmetric if for every deterministic the distribution of equals the distribution of . Properties The expectation of a circularly symmetric complex random vector is either zero or it is not defined. The pseudo-covariance matrix of a circularly symmetric complex random vector is zero. Proper complex random vectors A complex random vector is called proper if the following three conditions are all satisfied: (zero mean) (all components have finite variance) Two complex random vectors are called jointly proper is the composite random vector is proper. Properties A complex random vector is proper if, and only if, for all (deterministic) vectors the complex random variable is proper. Linear transformations of proper complex random vectors are proper, i.e. if is a proper random vectors with components and is a deterministic matrix, then the complex random vector is also proper. Every circularly symmetric complex random vector with finite variance of all its components is proper. There are proper complex random vectors that are not circularly symmetric. A real random vector is proper if and only if it is constant. Two jointly proper complex random vectors are uncorrelated if and only if their covariace matrix is zero, i.e. if . Cauchy-Schwarz inequality The Cauchy-Schwarz inequality for complex random vectors is . Characteristic function The characteristic function of a complex random vector with components is a function defined by: See also Complex normal distribution Complex random variable (scalar case) References Probability theory Randomness Algebra of random variables