Pseudononstationarity in the scaling exponents of finite-interval time series
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena.
Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary
stochastic process time series can yield anomalous time variation in the scaling exponents, suggestive of
nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations
is known for finite variance processes to vary as 1/N as N→
infinity for certain statistical estimators;
however, the convergence to this behavior will depend on the details of the process, and may be slow.We study
the variation in the scaling of second-order moments of the time-series increments with N for a variety of
synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable
N, one is far from this 1/N limiting behavior. We propose a semiempirical estimate for the minimum N
needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare
these with some “real world” time series.