Strong laws of large numbers for random walks associated with a class of one-dimensional convolution structures (Q1207680)

Fix \(\alpha\in[0,1[\). Let \(a_ n,c_ n>0\), \(b_ n\geq 0\) be real numbers satisfying \(a_ n+b_ n+c_ n=1\) for \(n\in\mathbb{N}\). Assume that \(\lim_{n\to\infty} n^ \alpha(a_ n-c_ n)=:\rho>0\) and \(\lim_{n\to\infty} b_ n=:\beta<1\). Let \((S_ n)_{n\in\mathbb{N}_ 0}\) be a birth-and-death random walk on the set \(\mathbb{N}_ 0\) of nonnegative integers, i.e. a Markov chain with \(S_ 0=0\) having the transition probability \(P(S_{n+1}=l\mid S_ n=k)=\mu_{l,k}\) where \(\mu_{k+1,k}=a_ k\), \(\mu_{k,k}=b_ k\), \(\mu_{k-1,k}=c_ k\) for \(k\in\mathbb{N}\), \(\mu_{1,0}=1\), and \(\mu_{l,k}=0\) otherwise. It is shown that \(\lim_{n\to\infty} S_ n/n^{1/(1+\alpha)}=(1-\alpha)\rho(1- \beta)^{-\alpha/(1+\alpha)}\) almost surely where almost sure bounds for the rate of convergence are given. This paper also contains strong laws of large numbers of this type for random walks on polynomial hypergroups defined via the coefficients \(a_ n\), \(b_ n\), \(c_ n\) above. Moreover, these laws are transferred to Sturm-Liouville hypergroups on \(\mathbb{R}_ +\) which admit an asymptotic behavior analog to the discrete case above. The limit cases \(\alpha=0,1\) have been studied earlier by \textit{H. Zeuner} [Math. Ann. 283, No. 4, 657-678 (1989; Zbl 0646.60009)] and the author [J. Theor. Probab. 3, No. 2, 245-266 (1990; Zbl 0719.60010)]. We mention that for \(\alpha=1\) the results are of a completely different kind.