We present examples of $p$-sub-exponential random variables for any positive $p$. We prove two types of concentration of standard $p$-norms ($2$-norm is the Euclidean norm) of random vectors with independent $p$-sub-exponential coordinates around the Lebesgue $L^p$-norms of these $p$-norms of random vectors. In the first case $p\ge 1$, our estimates depend on the dimension $n$ of random vectors. But in the second one for $p\ge 2$, with an additional assumption, we get an estimate that does not depend on $n$. In other words, we generalize some know concentration results in the Euclidean case to cases of the $p$-norms of random vectors with independent $p$-sub-exponential coordinates.