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The easiest condition would be a bound on $\sup_i \mathbb{E} X_i^6$, which would allow you to apply the Berry–Esseen theorem. More generally, if for some $0<\varepsilon < 2$ you have a uniform bound on $\mathbb{E} |X_i|^{4+\varepsilon}$, then you can apply the Lyapunov condition (which, as I noted above, follows from Lindeberg's condition by the Markov/Chebyshev inequality).