Invariance principles for random walks on hypergroups on \(\mathbb{R}_ +\) and \(\mathbb{N}\). (English) Zbl 0793.60006
Summary: Let \((X_ n:n\in\mathbb{N})\) be i.i.d. with finite variance and values in a hypergroup \(K:=\mathbb{R}_ +\) or \(\mathbb{N}\) and \(\Lambda\sum^ n_{j=1}X_ j\) be the randomized sum of these random variables. It is shown that the processes \((\Lambda\sum^{\lfloor nt\rfloor}_{j=1}(X_ j/\sqrt n):t\geq 0)\) converge in distribution to a Gaussian process in the case \(K=\mathbb{R}_ +\), that the processes \((1/\sqrt n\cdot\Lambda\sum^{\lfloor nt\rfloor}_{j=1}X_ j:t\geq 0)\) converge towards a Bessel process on \(\mathbb{R}_ +\) in the case of polynomial growth of the hypergroup \(K=\mathbb{R}_ +\) or \(\mathbb{N}\), and that in the case of exponential growth \((1/ \sqrt n \cdot (\Lambda \sum^{\lfloor nt \rfloor}_{j=1}X_ j- \lfloor nt \rfloor E_ *(X_ 1)):t \geq 0)\) converges towards a Brownian motion as \(n \to \infty\).
MSC:
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
Keywords:
invariance principle; Sturm-Liouville hypergroup; polynomial hypergroup; central limit theorem; Bessel process; Brownian motionReferences:
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