Your exact question seems to be answered by Lenny Baum and Melvin Katz in their 1963 paper. (convergence rates in the law of large numbers)

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