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We discuss a model for long memory and persistence in time series that amounts to harmonically weighting short memory processes, ∑jxt−j/(j+1). A non‐standard rate of convergence is required to establish a Gaussian functional central limit theorem. Theoretically, the harmonically weighted (HW) process displays less persistence and weaker memory than the classical competitor, fractional integration (FI) of order d. Still, we establish that a test rejects the null hypothesis of d = 0 if the process is HW. Similarly, a bias approximation shows that estimators of d will fail to distinguish between HW and FI given realistic sample sizes. The difficulties to disentangle HW and FI are illustrated experimentally and with USA inflation data. [ABSTRACT FROM AUTHOR]