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We use a multivariate version of Stein's method to establish a quantitativeLindeberg CLT for the Fourier transforms of random $N$-vectors. We achieve thisby deducing a specific integral representation for the Hessian matrix of asolution to the Stein equation with test function $e_t(x) = \exp(- i\sum_{k=1}^N t_k x_k)$, where $t,x \in \mathbb{R}^N$.

Stein's method and a quantitative Lindeberg CLT for the Fourier transforms of random vectors