How complete random permutations affect the dependence structure of stationary sequences with long-range dependence

A stationary time series is said to be long-range dependent (LRD) if its autocovariance function decays as a power of the lag, in such a way that the sum (over all lags) of the autocovariances diverges. The asymptotic rate of decay is determined by a parameter H, called the Hurst parameter. The time series is said to be short-range dependent (H = 1/2) if the sum converges. It is commonly believed that a random permutation of a sequence maintains the marginal distribution of each element but destroys the dependence, and in particular, that a random permutation of an LRD sequence creates a new sequence whose estimate of Hurst parameter H is close to 1/2. This paper provides a theoretical basis for investigating these claims. In reality, a complete random permutation does not destroy the covariances, but merely equalizes them. The common value of the equalized covariances depends on the length N of the original sequence and it decreases to 0 as N → ∞. Using the periodogram method, we explain why one is led to think, mistakenly, that the randomized sequence yields an "estimated H close to 1/2."