Limit theorems with random sample size for generalized domains of semistable attraction
Random central limit theorems, i.e. weak limits for scaled and normed versions of \(S_{T_n}\), have been established under a variety of conditions and settings, and this paper generalizes the randomized central limit theorem to generalized domains of multivariate semistable attraction (i.e. \(k_{n+1}/k_n\rightarrow c>1\)). The proofs employ three steps as in the paper of \textit{A. Rényi} [Acta Math. Acad. Sci. Hungar. 11, 97-102 (1960; Zbl 0091.14205)]; for \(T_n=\lfloor nD\rfloor\), then for \(T_n/n\rightarrow D\) in probability, and finally for arbitrary \(D>0\). The Anscombe condition for the third step is verified in Theorem 2.2. Theorem 2.4 states that under the above conditions \(A_n(S_{T_n}-({T_n}/{k_n})a_n)\) converges weakly to a distribution given by \(\int_0^\infty \mu^t d\rho(t)\).
scientific article; zbMATH DE number 1897042