Random Walk Weakly Attracted to a Wall

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We consider a random walk Xn in ℤ+, starting at X0=x≥0, with transition probabilitiesand Xn+1=1 whenever Xn=0. We prove 𝔼Xn ∼ const. n1-δ/2 as n↗∞ when δ ∈ (1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.

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