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In this paper, we investigate how close a fractional process can be to a random walk with drift in terms of the sample path. Given the innovation sequence, we calculate the distance to the closest random walk with drift in the sum of squares sense. We also derive the expected distance between the processes under the assumption of white noise normal innovations. A local approximation formula for this distance is given in terms of the sample size, showing that it increases with the sample size more rapidly than the square of the number of observations. Two empirical examples illustrate the results.

How Close Is a Fractional Process to a Random Walk with Drift?