Random walk weakly attracted to a wall
Abstract
We consider a random walk Xn in Z+, starting at X0=x>= 0, with transition probabilities P(Xn+1=Xn+1|Xn=y>=1)=1/2-δ/(4y+2δ) P(Xn+1=Xn+1|Xn=y>=1)=1/2+δ/(4y+2δ) and Xn+1=1 whenever Xn=0. We prove that the expectation value of Xn behaves like n1-(δ/2) as n goes to infinity when δ is in the range (1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.
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