  Let $X_1, \ldots, X_n\in\mathbb{R}^p$ be i.i.d. random vectors. We aim to
perform simultaneous inference for the mean vector $\mathbb{E} (X_i)$ with
finite polynomial moments and an ultra high dimension. Our approach is based on
the truncated sample mean vector. A Gaussian approximation result is derived
for the latter under the very mild finite polynomial ($(2+\theta)$-th) moment
condition and the dimension $p$ can be allowed to grow exponentially with the
sample size $n$. Based on this result, we propose an innovative resampling
method to construct simultaneous confidence intervals for mean vectors.
