Suppose we have a time series $x_t=\sin(0.02\pi t)$. Although this time series is totally deterministic, we can treat it as one realization of a proto/quasi/pseudo-stochastic process and estimate the distribution of $P(x_t)$.
Similarly, we can have time-lagged version of the sine time series $x_{t-\tau}=\sin[0.02\pi(t-\tau)]$ and its distribution is $P(x_{t-\tau})$.
Now I want to look at the joint-distribution $P(x_t,x_{t-\tau})$. Kernel density estimation with Silverman's rule will give something like (here $\tau=10$)
My Question is: what is the analytic expression of $P(x_t,x_{t-\tau})$?
Edit 1: $t$ is treated as uniformly distributed.