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The original motivation:

Let's say, we have two particles, doing random one motion on an infinite 1-D lattice $\mathbb{Z}$, and we are interested in the autocorrelation of the trajectory of each particle. we can do this easily, everything is known.

But, let's work in another space, $\mathbb{S}^1$ where there are finite lattice points (a circle with some number of lattice points on it). Mathematically speaking, the calculation of the autocorrelation of the trajectory of each particle is also easy, but in practice, to implement a circular lattice space, what we would do is that we first look at a bounded lattice $[-L, L]$, and allow particles to jump from one side to another should they want to go out of these boundaries.

This poses a practical problem; if we use the usual equations when we try to calculate the autocorrelations, the results will be wrong for the particles who made a jump at least once.

Question(s):

So, to pose a well-defined problem, how can I go about to make statistical calculations, such as calculating the autocorrelation of the trajectory of a particle who moves on a circular lattice?

But, in general, how to make any statistics on a space that is not Euclidean?

whuber
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    One solution is to use the unit circle in the Complex plane, mapping each integer $n$ to $\exp\left(2\pi n\,i/(2L+1)\right),$ and computing the autocorrelation function of those values. For other spaces, the solution depends on the geometry and why you are going through this exercise. Note, though, that any correlation can be computed in terms of distances between values, as described at https://stats.stackexchange.com/a/18200/919, providing a clear way to generalize to any geometry based on a metric. – whuber Jul 04 '20 at 15:02
  • @whuber thanks for a solution for that particular problem (who said it is an "exercise" by the way? :)) – Our Jul 04 '20 at 17:58

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