When is the probability distribution of the data important?

The null-hypothesis in the significance tests for WT, XWT and WTC is normally distributed AR1 noise. The AR1 coefficient and process variance is chosen so that it best fits the observed data. It is therefore quite important that the data is close to normal and is reasonably well modeled by a Gaussian AR1 process. Otherwise we can trivially reject the null-hypothesis and the significance level calculated by the program is not appropriate. However, the Central Limit Theorem tells us that the distribution tends towards normality as we convolute with longer and longer wavelets (in the absence of long-range persistence). This means that the data distribution is only really important on the shortest scales. So, if we are primarily looking at longer scales we do not need to worry so much about the distribution. However, for the WT and XWT the color of the noise is very important and a very non-normal distribution will affect the performance of the ar1 estimators (ar1.m & ar1nv.m). The WTC is relatively insensitive to the colour of the noise in the significance test (see next question).