Moment inequalities and the central limit theorem for integrals of random fields with mixing

Let $X_u$, $u\in R^q$ be a weakly dependent random field, $EX_u=0$, let $\mu$ be the Lebesque measure in $R^q$, let $V_n$ be an increasing system of subsets in $R^q$ and let $\zeta_n=(\mu(V_n))^{-1/2}\int_{V_n}X_n\,du$. One obtains a central limit theorem for $\zeta_n$ and estimates for the moments $E|\zeta_n|^t$, $t\ge2$.