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 approximately 1N 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 approximately 1N 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.