Selective Limit Theorems for Random Walks on Parabolic Biangle and Triangle Hypergroups

Abstract

Let K be respectively the parabolic biangle and the triangle in ${\mathbb{R}}^2$ and $\left( {\alpha \left( p \right)} \right)_{p \in {\mathbb{N}}}$ be a sequence in [0, +∞[ such that lim_p→∞ α(p)=+∞. According to Koornwinder and Schwartz,⁽⁷⁾ for each $p \in {\mathbb{N}}$ there exist a convolution structure (*_α(p)) such that (K, *_α(p)) is a commutative hypergroup. Consider now a random walk $\left( {X_j^{{\alpha }\left( p \right)} } \right)_{j \in {\mathbb{N}}}$ on (K, *_α(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables $\left( {X_j^{{\alpha }\left( p \right)} } \right)_{p \in {\mathbb{N}}}$ on K admit a selective limit theorems. The proofs depend on limit relations between the characters of these hypergroups and Laguerre polynomials that we give in this work.