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

Let K be respectively the parabolic biangle and the triangle in R^2 and (alpha(p))p (element of) N be a sequence in [0, +(infinity) [ such that limp(longrightarrow)(infinity) (alpha)(p)=+(infinity). According to Koornwinder and Schwartz,(7) for each p (element of) N there exist a convolution structure (*(alpha) (p)) such that (K, *(alpha)(p)) is a commutative hypergroup. Consider now a random walk (X alpha(p) j ) j (element of) N on (K, *(alpha)(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables (X alpha(p) j(p) ) p (element of) 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.