I want to check if a time series is (a) random or not (b) independent. For these I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future values.
If random variables are independent then there should be no correlation between them. The reverse is not true.
I am learning what techniques are there to show if a chaotic nonlinear dynamical system from which the time series is obtained can be a good generator of PRBS. I am following the tests given in paper : A New Pseudo-Random Number Generator Based on Two Chaotic Maps
Question : In general, for nonlinear time series, Lyapunov exponents tells us if the system is chaotic or not. But, for my case, I have to transform the values to +1/-1. In this case, would finding Lyapunov exponents be valid?
Estimating the Lyapunov exponents from a time series inherently requires to look at trajectories that have a small distance at some point (to see how they diverge from each other). If your time series is symbolised as you describe, the minimum distance two trajectories can have is 2, which is close to the diameter of the attractor. Hence, unless you do a high-dimensional embedding, you cannot even reasonably calculate the Lyapunov exponent.
I am not familiar with Matlab, but apparently it rescaled your plot in a weird manner. The auto- and cross-correlation should not obtain absolute values larger than one. Considering this and the symmetry of the autocorrelation plot, I would presume that the peak in the autocorrelation is actually located at 0, where there is no delay and you are looking at the correlation of the data with itself, which is always 1.
It looks as if there is a non-vanishing auto-correlation outside this, and your time series is hence correlated.