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An upper bound is derived for the mean-square error involved when a non-band-limited, wide-sense stationary random process (possessing an integrable power spectral density) is approximated by a cardinal series expansion of the form sinc , a sampling expansion based on the choice of some nominal bandwidth . It is proved that where sinc , and is the power spectral density for . Further, the constant is shown to be the best possible one if a bound of this type (involving the power contained in the frequency region lying outside the arbitrarily chosen band) is to hold uniformly in . Possible reductions of the multiplicative constant as a function of are also discussed, and a formula is given for the optimal value of this constant.