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We show that if the graph of an analytic function in the unit disk$D$is not complete pluripolar in$C$^{2}then the projection of its pluripolar hull contains a fine neighborhood of a point $p\in\partial\mathbf{D}$ . Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function$f$in$D$extends to a function ℱ which is defined on a fine neighborhood of a point $p\in\partial\mathbf{D}$ and is finely analytic at$p$then the pluripolar hull of the graph of$f$contains the graph of ℱ over a smaller fine neighborhood of$p$. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them$C$^{∞}functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.

Large-scale multiple testing with correlated and heavy-tailed data arises in a wide range of research areas from genomics, medical imaging to finance. Conventional methods for estimating the false discovery proportion (FDP) often ignore the effect of heavy-tailedness and the dependence structure among test statistics, and thus may lead to inefficient or even inconsistent estimation. Also, the assumption of joint normality is often imposed, which is too stringent for many applications. To address these challenges, in this paper we propose a factoradjusted robust procedure for large-scale simultaneous inference with control of the false discovery proportion. We demonstrate that robust factor adjustments are extremely important in both improving the power of the tests and controlling FDP. We identify general conditions under which the proposed method produces consistent estimate of the FDP. As a byproduct that is of independent interest, we establish an exponential-type deviation inequality for a robust U-type covariance estimator under the spectral norm. Extensive numerical experiments demonstrate the advantage of the proposed method over several state-of-the-art methods especially when the data are generated from heavy-tailed distributions. Our proposed procedures are implemented in the R-package FarmTest. Supported by NSF Grants DMS-1662139, DMS-1712591, and NIH Grant R01-GM072611-12.