The significance of trends in long-term correlated records

We study the distribution $P(x;\alpha,L)$ of the relative trend $x$ in long-term correlated records of length $L$ that are characterized by a Hurst-exponent $\alpha$ between 0.5 and 1.5 obtained by DFA2. The relative trend $x$ is the ratio between the strength of the trend $\Delta$ in the record measured by linear regression, and the standard deviation $\sigma$ 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 \cite{Lennartz2011} we show explicitely that $P$ follows the student-t distribution $P\propto [1+(x/a)^2/l]^{-(l+1)/2}$, where the scaling parameter $a$ depends on both $L$ and $\alpha$, while the effective length $l$ depends, for $\alpha$ below 1.15, only on the record length $L$. From $P$ we can derive an analytical expression for the trend significance $S(x;\alpha, L)=\int_{-x}^x P(x';\alpha,L)dx'$ and the border lines of the $95\%$ percent 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.