In this paper we present a study of the problem of approximating the
expectations of functions of statistics in independent and
dependent random variables in terms of the expectations of
functions of the component random variables. We present results
providing sharp analogues of the Burkholder--Rosenthal inequalities
and related estimates for the expectations of functions of sums of
dependent nonnegative r.v.'s and conditionally symmetric
martingale differences with bounded conditional moments as well as
for sums of multilinear forms. Among others, we obtain the
following sharp inequalities: $E(\sum_{k=1}^n X_k)^t\le 2 \max
(\sum_{k=1}^n EX_k^t, (\sum_{k=1}^n a_k)^t)$ for all nonnegative
r.v.'s $X_1, \ldots, X_n$ with $E(X_k\mid X_1, \ldots, X_{k-1})\le a_k$,
$EX_k^t<\infty$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$; $E(\sum_{k=1}^n
X_k)^t\le E\theta^t(1) \max (\sum_{k=1}^n b_k, (\sum_{k=1}^n
a_k^s)^{t/s})$ for all nonnegative r.v.'s $X_1, \ldots, X_n$ with
$E(X_k^s\mid X_1, \ldots, X_{k-1})\le a_k^s$, $E(X_k^t\mid X_1, \ldots,
X_{k-1})\le b_k$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$, $0#x003C;s\le t-1$ or $t\ge
2$, $0#x003C;s\le 1$, where $\theta(1)$ is a Poisson random variable
with parameter 1. As applications, new decoupling inequalities for
sums of multilinear forms are presented and sharp
Khintchine--Marcinkiewicz--Zygmund inequalities for generalized
moving averages are obtained. The results can also be used in the
study of a wide class of nonlinear statistics connected to
problems of long-range dependence and in an econometric setup, in
particular, in stabilization policy problems and in the study of
properties of moving average and autocorrelation processes. The
results are based on the iteration of a series of key lemmas that
capture the essential extremal properties of the moments of the
statistics involved.