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 .

Reminder of the notation introduced in Fort, Moulines (2003) p.12.

Let , a sequence of random variables and a sequence of nonzero real numbers. We write if is bounded in .

Let and a sequence of random variables such that with . Then,

Remark: The assumptions of the theorem hold as soon as for all and with .

Remark: For , look at Theorem 2.19 of Hall, Heyde Martingal Limit Theory and its application.

Let be a compact subset of with , a measurable space, a random variable taking its values on , and a measurable function defined on . Define on the function , and for all and , the Monte-Carlo average where and are i.i.d. random variables with .

Assume that:

1. is continuous on ,
2. there exists a measurable function such that a.s., for all , with ,
3. .

Then,

Remark. If is continuous with respect to the first variable , by Lebesgue's dominated convergence theorem under assumption 2. the function is continuous (i.e. assumption 1. is verified).