Consider a random walk on the number line where, at each step, the position (call it x) may change by +1 (to the right) or −1 (to the left) with probabilities:

p(go right)= 1/2+1/2*(x/(1+abs(x)) for abs(x)below 8

p(go right)=0 for x=8

p(go right)=1 for x= -8

p(go left)= 1- p(go right)

If 10000 steps of a walk are taken into account: :

a. What is the distribution of distances x appearing in the chain 01212343234543 … etc ... :

b: Explain why the said distribution is in the long run independent of the initial state.