Geodesic
Moment inequalities and the central limit theorem for integrals of random fields with mixing
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 39-47

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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 $E|\zeta_n|^t$, $t\ge2$. 
@article{ZNSL_1985_142_a2,
     author = {V. V. Gorodetskii},
     title = {Moment inequalities and the central limit theorem for integrals of random fields with mixing},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {39--47},
     publisher = {mathdoc},
     volume = {142},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a2/}
}
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V. V. Gorodetskii. Moment inequalities and the central limit theorem for integrals of random fields with mixing. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 39-47. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a2/