Quote from nononsense:
To discuss this, you will have to describe the way you generate your time series. I'll try to do this in a simplified manner.
(1) deterministic time series where the value at a particular time can be represented by a function;
(2) random or stochastic time series where a value at a particular time cannot be represented by such a function. In fact these random signals are generated through standardized processes well defined through their probabilistic properties. Popular ones are the 'Wiener-Levy Process', 'Poisson Process', 'Brownian Motion'.
'White Noise' can be defined by a (tricky) limiting process on the above but is commonly defined as having a flat power spectral density function yielding the delta function as autocorrelation function.
Passing this 'White Noise' random signal through a linear (even non-linear) filter generates a new signal, equally random, called 'colored noise'. The spectral density function is no longer flat and the autocorrelation function is no longer a delta function (spike). This makes this new but still perfectly random signal predictable in the sense that a filter can be found yielding the optimal approximation in some mathematical sense (minimum integral RMS error). Of course, the probabilistic parameters of the random signal have to satisfy the requirements of the theory. Wiener developed his filtering theory. Other more advanced methods exist, eg Kalman filters. These are linear filters. Some work has been done on non-linear filtering but this becomes rapidly involved.
When dealing with markets, it is useful to have an idea of these theories. This does not mean that these will lead to any useful result. In fact, most often one will remain far removed from satisfying the requirements for applying these theories.