Concentration of norms of random vectors with independent \(p\)-sub-exponential coordinates

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.