Abstract
Christian Houdre has kindly pointed us to a paper by A. Fuks, A. Joffe and J. Teugels, where their Theorem 5.3 is our Proposition 3 in the case $0 < \alpha < 1$.
Citation
David Mason. Joel Zinn. "Acknowledgment of Priority: When Does a Randomly Weighted Self-normalized Sum Converge in Distribution? (Elect. Comm. in Probab. 10 (2005), 70-81)." Electron. Commun. Probab. 10 297 - 297, 2005. https://doi.org/10.1214/ECP.v10-1170
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