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

Abstract 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 $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -norms of these p -norms of random vectors. In the first case $$p\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , our estimates depend on the dimension n of random vectors. But in the second one for $$p\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , with an additional assumption, we get an estimate that does not depend on n . In other words, we generalize some known concentration results in the Euclidean case to cases of the p -norms of random vectors with independent p -sub-exponential coordinates.