On powers of likelihood functions of random walks on ℤͩ

Let {Xi}i≥1 be independent, identically distributed random vectors in ℤd,d≥1. Let LLn(x)≡ℙ(Sn=x),n≥1,x∈ℤd, be the likelihood function for Sn=∑i=1nXi. For integers j≥2 and n≥1, let an(j)≡∑x∈ℤd(Ln(x))j. We show that if X1-X2 has a nondegenerate aperiodic distribution in ℤd and 𝔼(∥X1∥2)>∞, then limn→∞n(j-1)d∕2an(j)≡a(j,d) exists and 0

Keywords:

• Combinatorics
• Mathematics
• Random walk

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