In this paper: A Practical Guide to Wavelet Analysis, I read that "The wavelet transform can be used to analyze time series that contain nonstationary power at many different frequencies."
I also read the paper The wavelet transform, time-frequency localization and signal analysis by I. Daubechies but I couldn't figure out this sentence.
Anyway, my question is: does the wavelet transform (e.g. DWT) require the stationarity of time series or I can apply it also to non-stationary signals?
Yes, wavelets can be used on nonstationary time series.
A common wavelet used for this type of data is DTW (discrete wavelet transform). The same transform can be used with nonstationary data that has long memory.
There are several books and lots of papers that deal with wavelets and nonstationary data. I have used Gencay, Selcuk and Whitcher's text, published by Academic Press in 2002. For many years now.