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I am trying to obtain samples of a random walk on a circle that takes steps according to N(0, \sigma). To do this for a discrete random walk one calculates the cumulative sum of {+1,-1} coin tosses and then does mod the number of states on the circle at each step to obtain the current position. My question is: how to modify this for the continuous case?

Thanks in advance if anyone can help!

hawk.q852
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  • not sure, but does `cumsum(rnorm(n,0,sigma)) %% (2*pi)` do what you need? – Ben Bolker Dec 11 '20 at 15:27
  • What is Normally distributed in your example? The distance of the arc round the circle or the angle of the position relative to some fixed reference? Something else? Also, I would say your example of a discrete RW is just a specific example. It's not the only way of doing it. – Limey Dec 11 '20 at 15:31
  • @Limey I was thinking, it's the distance of the arc that should be Normal, but I'm just not sure how to draw samples of the walk in that case to show that the stationary distribution is Uniform. Btw, what's the other way to do a discrete RW? I've only heard of the one I mentioned above – hawk.q852 Dec 11 '20 at 16:28
  • @BenBolker yes, I think it does, thank you so much! – hawk.q852 Dec 11 '20 at 16:34

1 Answers1

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If the N(0,sigma) step distribution refers to the distribution of arc lengths around the circle, then

 cumsum(rnorm(n,0,sigma)) %% (2*pi)

should give a sample of the steps on the circle (in radians), starting from zero.

Ben Bolker
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