# Question about autocorrelation of time series

I have a time series $$S(t)$$ for $$t\in I$$ where $$I=[0,t_\text{max}]$$ is an interval. The time series is very regular, $$S(t) = \sin(t)$$, although my following question is independent of the explicit analytic form (if any) of the series.

I want to select two sub-intervals of $$I$$ of same length, for example $$I_0 = [t_0,t_1]$$ and $$I_1 = [t_1,2t_1-t_0]$$ for some $$t_0>0,t_1>t_0$$. Now, I consider the two following series

$$S_0 (t) = S(t) \,,\quad \text{for}\,\,t \in I_0\\ S_1 (t) = S(t_1-t_0+t) \,,\quad \text{for}\,\,t \in I_0$$

and I am interested in computing the correlation between $$S_0 (t)$$ and $$S_1 (t)$$

$$\text{corr}\left(S_0 (t),S_1 (t)\right)\,,\qquad (\text{Eq}.1)$$

Is this quantity somewhere defined in statistics? Does it have any statistical property?

• Is $t$ continuous or discrete time? The notation suggests it is continuous, but I just want to be sure. Apr 13 at 13:45
• @RichardHardy It is continuous for $S(t)=\sin(t)$ but eventually I want to consider discrete time Apr 13 at 16:31
• That is auto-correlation at lag $t_1-t_0$. See here. Apr 14 at 7:49