I'm working out the proof of Donsker's theorem given in Revuz and Yor, Continuous martingales and BM, (theorem 1.9, ch. XIII, p. 518).
At one certain point, they prove the following fact : for every sequence of bounded i.i.d. centered random variables with variance $\sigma=1$, for every $\lambda >0$ :
$$\limsup_{n \to \infty} \mathbf{P}(\max_{i \leqslant n} |S_i| > \lambda \sqrt{n} ) \leqslant \frac{3}{\lambda^4} $$
where $S_i = \xi_1 + ... + \xi_i$. This is done using classical martingale techniques ; the bounding hypothesis is used to show that $S_n$ is in $L^4$.
Then, to generalize to unbounded $\xi_i$, they state "by truncating and passing to the limit, it may be proved that this is still true if we remove the assumption that $\xi_k$ is bounded".
I can't find the way they use some truncation argument. You can't truncate the $\xi_k$, otherwise they might not be centered anymore, and I can't see any other thing to do. Did somebody actually found a way to "truncate then pass to the limit" ?
edit : maybe we could stop the martingale $S_n$.
Fix $R>0$ and define $$X'_i:=X_i\mathbf 1\{|X_i|\leqslant R \}-\mathbb E\left[X_i\mathbf 1\{|X_i|\leqslant R \}\right]\mbox{ and} $$ $$X''_i:=X_i\mathbf 1\{|X_i|\gt R \}-\mathbb E\left[X_i\mathbf 1\{|X_i|\gt R \}\right].$$ The sequence $\left(X'_i\right)_{i\geqslant 1}$ is bounded, centered and its variance is smaller than $ \mathbb E\left[X_1^2\mathbf 1\{|X_1|\gt R \}\right]$ which is it self bounded by $\mathbb E\left[X_1^2\right]\leqslant 1$. For the unbounded part, we use Doob's inequality to obtain $$\mathbf{P}\left(\max_{i \leqslant n} \left|\sum_{j=1}^i X_j'' \right| > \lambda \sqrt{n} \right)\leqslant \frac 2{\lambda^2}\mathbb E\left[{ X''_1} ^2\right] \leqslant 8\mathbb E\left[X_1^2\mathbb 1\{|X_1|\gt R\} \right].$$