In the book "Sums of Independent Random Variables" by Petrov the following lemma appears in page 109, in preparation to prove the Berry-Esseen inequality
Let $X_1,...,X_n$ be independent random variables, $\mathbb{E}[X_i]=0$ and $\mathbb{E}[|X_i^3|]<\infty$ for $1\leq i \leq n$. Denote $$\sigma_i^2 = \mathbb{E}[X_i^2],\ B_n = \sum\limits_{i=1}^n\sigma_i^2,\ L_n=B_n^{-3/2}\sum\limits_{i=1}^n\mathbb{E}[|X_i|^3]$$
Let $\varphi_n$ denote the characteristic function of the random variable $B_n^{-1/2}\sum\limits_{i=1}^nX_i$. Then $$|\varphi_n(t) - e^{-t^2/2}| \leq 16 L_n|t|^3e^{-t^2/2}$$ for $|t|\leq \frac{1}{4L_n}$.
In the statement above $e^{-t^2/2}$ is just the characteristic function of the standard Gaussian.
I was wondering if there is a higher dimensional analogue to this lemma. Specifically, if $\varphi_n$ is the characteristic function of a sum of (nice enough) independent random vectors with dimension $d$. What can we say about the difference $$|\varphi_n(\bar{x})-e^{-||\bar{x}||^2/2}|$$ for $\bar{x} \in \mathbb{R}^d$ with some restriction on the norm.
I would be happy to hear any related results. References would come in handy as well.
Thanks.