Convergence results for a normalized triangular array of symmetric random variables (Q1857375)

Let \(\{X_{n,j},n\geq 1,\;1\leq j\leq k_n\}\) be a triangular array of random variables. Suppose that, for each \(n\), the vector \((X_{n,1}, X_{n,2}, \dots, X_{n,k_n})\) is jointly symmetric, i.e., all the vectors of the form \((\varepsilon_1X_{n,1}, \varepsilon_2X_{n,2}, \dots,\varepsilon_{k_n} X_{n, k_n})\), where \(\varepsilon_j \in\{-1,1\}\), \(j=1,2, \dots,k_n\), have the same distribution. Set \(S_n=\sum^{k_n}_{j=1} X_{n,j}\) and \(V_n= (\sum^{k_n}_{j=1} X^2_{n,j})^{1/2}\). The author examines the weak limit behaviour of the normalized sums \(T_n=S_n V_n^{-1}\).