The difficulty of telling signal from noise when noise lingers

How many observations does it take to detect a fixed signal when the noise has long memory? For a sample-mean detector applied to a constant signal buried in fractionally integrated noise of order d∈0,1/2), I give a closed-form expression for the sample size required to reach a target power at a fixed size: Tq(d)=(τ+Φ−1(q))σcd/(δx)]2/(1−2d). The exponent 2/(1−2d) is the whole story: the required sample size is a power function of detection difficulty whose exponent diverges as d→1/2. A doubling of detection difficulty that costs four times the data under independence costs 210 times the data at d=0.4. A Monte Carlo confirms the formula delivers the target power and size.