Let be a random variable non-negative a.s. and an increasing differentiable function such that . Then,

Let , and real-valued random variables. Then, by convexity

Set . Let . We first establish an upper-bound for .

Let and define for all , and . Then, Let . The Chernoff bound and the independence of the by independence of the both provide Using the Taylor formula with the exponential function yields that the function defined on by is increasing, and together with for all , we deduce The fact that implies and thus . Combining with yields for all , Together with \eqref{eq:s_n_t_n} and \eqref{eq:t_n_only} this provides where . Note that implies that the are all equal to zero a.s., a situation where the inequality is trivially true, and we can thus assume . The argument of the exponential in \eqref{eq:s_n_step_1} is then minimized in at , with

With where , combining with \eqref{eq:s_n_step_1} yields Considering provides a similar inequality for , and using the fact that we deduce

Using the $g$-lemma with we deduce with the change of variables . The integral is finite iff , and we can choose to deduce the Rosenthal inequality: where only depends on . Finally, by Jensen inequality as , and by convexity,

which together with the Rosenthal inequality \eqref{eq:rosenthal} yields the Marcinkiewicz–Zygmund inequality: where .