¿Qué es Long memory, fractional differencing, Hurst exponent?

A stationary process has long memory if its autocorrelations decay like a power, $\rho(h) \sim ch^{2d-1}$ with $0 < d < \frac12$, so that they are not summable. Fractional differencing applies $(1 - L)^d = \sum_{j \ge 0}w_jL^j$ with $w_0 = 1$, $w_j = w_{j-1}(j - 1 - d)/j$; the ARFIMA$(0, d, 0)$ process is $(1 - L)^dX_t = \varepsilon_t$ (Granger and Joyeux, 1980; Hosking, 1981). The Hurst exponent is $H = d + \frac12$, measured by the growth of ranges or variances of partial sums like $n^H$ (Hurst, 1951).