Significance of trends in long-term correlated records.

We study the distribution P(x;α,L) of the relative trend x in long-term correlated records of length L that are characterized by a Hurst exponent α between 0.5 and 1.5. The relative trend x is the ratio between the strength of the trend Δ in the record measured by linear regression and the standard deviation σ around the regression line. We consider L between 400 and 2200, which is the typical length scale of monthly local and annual reconstructed global climate records. Extending previous work by Lennartz and Bunde [S. Lennartz and A. Bunde, Phys. Rev. E 84, 021129 (2011)] we show explicitly that x follows the Student's t distribution P∝[1+(x/a)2/l]-(l+1)/2, where the scaling parameter a depends on both L and α, while the effective length l depends, for α below 1.15, only on the record length L. From P we can derive an analytical expression for the trend significance S(x;α,L)=∫(-x)xP(x';α,L)dx' and the border lines of the 95% significance interval. We show that the results are nearly independent of the distribution of the data in the record, holding for Gaussian data as well as for highly skewed non-Gaussian data. For an application, we use our methodology to estimate the significance of central west Antarctic warming.