Hurst exponents larger than 1/2: long-range dependence or Levy flights? A tutorial

Kristoffer Rypdal
University of Tromsų, Norway

In the experimental study of turbulence we usually deal with discretely sampled data sets {X(n)}. When observations are made in one spatial location, the order parameter 'n' is time, and we deal with time series. An elementary result in statistics is that the mean of a sample consisting of N independent observations has a variance which is the variance of a single observation divided by N, i.e. var(X)/N. If the N observations are interpreted as successive increments (steps) in a random walk, the sum of increments Y(N)=X(n) (i.e. the position after N steps) is N times the sample mean. Thus the variance of "the position" is var(Y)=var(X) N, i.e. it grows linearly with time t=N dt. This situation is changed if the autocorrelation function for the increments has a fat tail so that its integral diverges. If the tail decays with increasing time as t-a, with exponent a<1, var(Y) -> t2-a for large t and the power spectrum has the form fa-1 for small f. In the limit a -> 1 we recover the same asymptotic dependence as for independent increments and a flat white noise spectrum, and for a -> 0 we approach a determinstic evolution and an 1/f spectrum.

All these considerations fail if the PDF for X(n) does not have finite variance, which is the case if the PDF has power-law tails p(X) -> X-(1+m), with 1< m< 2. In this case the sum Y(N) of independent increments converges to a stable (Levy) distribution with infinite variance, but the typical value of |Y(N)| (or the range R(N)) goes asymptotically like N1/m. Thus, if care is not taken to exclude the possibility of infinite variance of X(n), this behaviour of R(N) can be taken as indication of long-range dependence with a=2-2/m. The sample standard deviation S(N) converges to the standard deviation for X(n) when the latter exists. Otherwise S(N) -> N1/m-1/2. Thus, for independent increments the ratio R/S -> N1/2, i.e. the division by S removes the effect of fat tails on the asymptotic growth of the range, and a dependence R/S -> NH with Hurst exponent H > 1/2 should be due to long-range dependence.

If a turbulent field X is sampled at high frequency, the value of X(n) and X(n+1) will be very close, and hence highly correlated. Thus, on short time scales X(n) will not behave as independent increments, but perhaps rather as a sum of more or less independent increments, i.e. on this time scale {X(n)} behaves more like a position process of a random walk than an increment process. This suggests that we should look at the statistics (PDF, autocorrelation, R/S) for the increments of the process {X(n)}, i.e. for the process {Z(n)}, where Z(n)=X(n+1)-X(n). H > 1/2 for {Z(n)} indicates dependence among increments. In particular this is true if H is different on short and long time scales. If the Z(n) were independent H should not depend on the time lag, and the result should be the same if the time series is shuffled.

We discuss models that can describe the turbulent field as the position process in a random walk with damping included to prevent the field from diffusing to infinity for large times. We also present analyses of time-series from a plasma experiment, which illustrate some of the complications that often appear in real world data, such as long-range dependence due to more or less coherent oscillations.