This note complements a recent article by Thomas (1972). We prove a conjecture of Thomas which, in a way, makes his results complete. The proof uses the fact that for the sequences of random variables involved, the weak convergence property is conserved when taking certain transformations of the random variables.
"On Random Indices and Limit Distributions." Ann. Probab. 2 (1) 181 - 181, February, 1974. https://doi.org/10.1214/aop/1176996767