The main results are functional central limit theorems for superpositions of randomly selected partial sums in which the random variables being summed are independent and have distributions in the domain of attraction of stable laws. These results extend those of Tucker and Sreehari concerning when convolutions of distributions are attracted to stable laws. Other functional central limit theorems are presented for more general sums. The results herein extend the central limit theory for additive processes on Markov chains.
"Weak Convergence of Superpositions of Randomly Selected Partial Sums." Ann. Probab. 1 (6) 1044 - 1056, December, 1973. https://doi.org/10.1214/aop/1176996810