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Certain isospectral classes of random processes and certain linear representations of these processes are considered. It is shown that the Gaussian member of each class is the only one having the desirable property of being representable as a linear combination of statistically independent components. For example, among all strictly stationary, band-limited, white-noise processes, only the Gaussian process has mutually independent Nyquist samples. Characterizations among classes wider than isospectral classes are also obtained. For example, let x(t) have a rational spectral density with Karhunen-Loève (K-L) coefficients for the interval . It is argued that if the coefficients are mutually independent for each of a sequence of intervals with , then is Gaussian. This conclusion makes use of a more general characterization of Gaussian processes that is obtained using a characterization of the Gaussian distribution among infinitely divisible distributions. It also uses a conjecture about the behavior of the K-L representation of a known function as . Finally, certain non-Gaussian processes defined as sums of a random number of random pulses are considered. Necessary and sufficient conditions for the independence of linear functionals of this process are obtained.