Exponential Rosenthal and Marcinkiewicz–Zygmund inequalities

We extend the Rosenthal inequalities and the Marcinkiewicz–Zygmund inequalities to some exponential Orlicz spaces.The Rosenthal inequalities and the Marcinkiewicz–Zygmund inequalities are fundamental estimates on the moment of random variables on Lebesgue spaces. The proofs of the Rosenthal inequalities and the Marcinkiewicz–Zygmund inequalities on the exponential Orlicz spaces rely on two results from theory of function spaces and probability theory. The first one is an extrapolation property of the exponential Orlicz spaces. This property guarantees that the norms of some exponential Orlicz spaces can be obtained by taking the supremum over the weighted norms of Lebesgue spaces. The second one is the sharp estimates for the constants involved in the Rosenthal inequalities and the Marcinkiewicz–Zygmund inequalities on Lebesgue spaces. Our results are applications of the extrapolation property of the exponential Orlicz spaces and the sharp estimates for the constants involved in the Rosenthal inequalities and the Marcinkiewicz–Zygmund inequalities on Lebesgue spaces. In addition, the sharp estimates for the constants involved in the Rosenthal inequalities and the Marcinkiewicz–Zygmund inequalities on Lebesgue spaces provide not only some sharpened inequalities in probability, but also yield some substantial contributions on extending those probability inequalities to the exponential Orlicz spaces.