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The role of weak dependence between observed samples is investigated in the detection of time varying signals in noise, with the goal of establishing quantitative conditions for when the dependency can be ignored. A result is presented that allows bounding the variation in false-alarm rate and detection probability induced by ignoring the dependency. This result is applied to the case of stationary Gaussian noise, and it is shown that the dependency can be ignored if the noise autocorrelation decreases sufficiently fast. In fact, a bound on the variation in false-alarm rate and detection probability is linked to the rate of descent of the noise autocorrelation through an expression that can be easily evaluated.