Your exact question seems to be answered by Lenny Baum and Melvin Katz:
- Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108-123, https://doi.org/10.1090/S0002-9947-1965-0198524-1
Your exact question seems to be answered by Lenny Baum and Melvin Katz: