I recommend that you have a look at the beautiful survey paper

J.-P. Kahane: A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. 29(1997), 257-279

You can read more about this in the excellent survey

J.-P. Kahane: A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. 29(1997), 257-279

In particular you can learn from this survey that the phenomenon you mentioned is rather typical. It's definitely worth having a look at it.