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Random Beta Walk

Here we will introduce a random walk on $ (0,1)$ in which the next step $ X_{t+1}$ is determined by the beta distribution with parameter $ \alpha = \max(\delta+\theta X_t,1)$ and $ \beta = \max(\delta + \theta (1-X_t),1)$. Later we use this random walk as a proposal Markov chain on $ (0,1)$. A smaller $ \delta$ keeps a sample path closer to either of the boundary. The larger the value $ \theta$ is the smaller the move of each step becomes Thus, $ 0 < \delta < 1$ will change the shape of stationary distribution of the random walk, and $ \theta > 0$ will influence the speed of convergence of random walk.

Explore it. Download bwalk.r, and see how a sample path of the beta random walk looks for a different choice of $ \delta$ and $ \theta$.

> source("bwalk.r")
> sample.path = rwalk(move=bmove, trajectory=T, delta=0.8, theta=20)
> plot(sample.path, type="l", xlab="time", ylab="state", main="Beta random walk")
A long run behavior can be observed from the histogram of $ X_t$ from the end of runs repeatedly. Change the running time, and see if the distribution of $ X_t$ is different. Also obtain the histogram of $ X_t$ for a different choice of $ \delta$ and $ \theta$.
> sample.data = rwalk(move=bmove, run.time=100, delta=0.8, theta=20, sample.size=500)
> hist(sample.data, freq=F, breaks=seq(0,1,by=0.05), col='red')

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