Inequalities for the probabilities of large deviations in the multi-dimensional case

Let $X_1,\dots,X_n$ be independent random vectors with zero mean vectors. Let
\begin{gather*} \Lambda_i=\mathbf E|X_i|^2,\quad M_i=\mathbf E|X_i|^3,\quad\Lambda=\frac1n\sum_{i=1}^n\Lambda_i,\quad M=\frac1n\sum_{i=1}^nM_i, \\ Y_n=\frac1{\sqrt n}(X_1+\dots+X_n) \end{gather*}
We prove the following
Theorem. There exist absolute constants $K_1$ and $K_2$ such that for any $x>0$
$$ \mathbf P(|Y_n|\ge x)\le4\exp(-K_1x^2/\Lambda)+K_2M/\sqrt nx^3 $$