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 $P(x_t)$. For example, if kernel density kernel is used, we will get something like the following image Obviously the estimations at the two ends of the interval [-1,1] are messed up since a sine series can never go outside this range. The image just gives you an idea of what I want.
Question: What is the analytical expression of $P(x_t)$?
Edit 1: The bandwidth I used is the Silverman's rule: B.W. Silverman, “Density Estimation for Statistics and Data Analysis”, Vol. 26, Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1986.