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May 23

How to find the stationary distribution of a random walk on real numbers?

Start at the point $(1/2,1/2)$ . At each time step, choose a point uniformly at random from $(0,0),(0,1),(1,0),(1,1),(1/2,1/2)$ , and then move half way from the current position to the chosen point.

In other words, at each time step transition from $(x,y)$ to one of ${(x/2,y/2),(x/2,(y+1)/2),((x+1)/2,y/2),((x+1)/2,(y+1)/2),((x+1/2)/2,(y+1/2)/2)}$ .

After $n$ time steps measure the distance from the starting position.

What is the distribution of distances as $n\rightarrow \infty$ ? What is the expected distance?

I understand how to find stationary distributions for finite Markov chains, but this is a bit beyond me. What topics would be needed to answer this, or is there a general approach to answer questions of this form? 98.190.129.147 (talk) 23:41, 23 May 2019 (UTC)[reply]

What you're describing is similar to the chaos game. There are several closely related examples there, but not quite the one you're describing exactly. I suspect it might be worth it to just write a program to see if there's anything interesting as a result (often you get something self-similar from such a process). I'm not sure how to analyze a description like that systematically, but there might be some better refs at that article. –Deacon Vorbis (carbon • videos) 00:24, 25 May 2019 (UTC)[reply]

I did a plot of the cumulative distribution function for the x-variable alone; it's not that hard to compute for values of the form p/2ⁿ where n is small. Judging from the roughish appearance of the graph the cdf is fractal in nature, something like a smoothed version of the Cantor function. It would follow that the distribution has no density function in the normal sense. The average position is obviously (1/2, 1/2), and from that the average displacement is 0, but the average distance is a more difficult calculation even you just look at horizontal or vertical distance. Given a closed form for the distribution is unlikely, I think your best bet is to use a Monte Carlo method to get an estimate for something like the average distance. --RDBury (talk) 13:37, 25 May 2019 (UTC)[reply]