Simple proofs of Rosenthal inequalities for linear forms of independent random variables and its generalization to quadratic forms

The Rosenthal inequality (without specifying the dependence on p) was obtained by Rosenthal in 1970, [3]. An extensive literature is devoted to the refinements and various generalizations of the Rosenthal inequality. In this note, Theorem 1 refines the Rosenthal-type inequality that was obtained in the paper [2] of Pinelis (1994) and gives its elementary proof. Exact inequality constants are given. Theorem 2 represents the generalization of the inequality to quadratic forms. This result without specifying the constants follows from the paper [1] of Gine, Latala, Zinn (2000), but we give exact inequality constants. The method of proof of both theorems uses the recurrent estimation of moments and goes back to the ideas of Stein to some extent. More precisely, to his work in 1972. Both inequalities are widely used, for example, in the proof of local limit theorems for the spectrum of random matrices. The variant of the inequality of Theorem 2 was proved in asimilar way in the joint work [6] of the author and F. Goetze (2016). Proposed approach can be extended to the estimations of moments of U -statistics and more general statistics.