This paper studies various distributional properties of the Rosenblatt distribution. We begin by describing a technique for computing the cumulants. We then study the expansion of the Rosenblatt distribution in terms of shifted chi-squared distributions. We derive the coefficients of this expansion and use these to obtain the L\'{e}vy-Khintchine formula and derive asymptotic properties of the L\'{e}vy measure. This allows us to compute the cumulants, moments, coefficients in the chi-square expansion and the density and cumulative distribution functions of the Rosenblatt distribution with a high degree of precision. Tables are provided and software written to implement the methods described here is freely available by request from the authors.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ421 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Let {D(s), s ≥ 0} be a Lévy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that D(0) = 0. We study the first-hitting time of the process D, namely, the process E(t) = inf{s : D(s) > t}, t ≥ 0. The process E is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the n-time tail distribution function P [E(t1) > s1, . . . , E(tn) > sn], and show that this PDE has a unique solution given natural boundary conditions. This PDE can be used to derive all n-time moments of the process E. * Revised †

Let {D(s), s ≥ 0} be a non-decreasing Lévy process. The first-hitting time process {E(t) t ≥ 0}(which is sometimes referred to as an inverse subordinator) defined by E(t) = inf{s : D(s) > t} is a process which has arisen in many applications. Of particular interest is the mean first-hitting timeThis function characterizes all finite-dimensional distributions of the process E. The function U can be calculated by inverting the Laplace transform of the function e U (λ) = (λφ(λ)) −1 ,where φ is the Lévy exponent of the subordinator D. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.

In this work deep convolutional neural networks (CNNs) are shown to be an effective model for fusing heterogeneous geospatial data to create radar-like analyses of precipitation intensity (i.e., synthetic radar). The CNN trained in this work has a directed acyclic graph (DAG) structure that takes inputs from multiple data sources with varying spatial resolutions. These data sources include geostationary satellite (1-km visible and four 4-km infrared bands), lightning flash density from Earth Network’s Total Lightning Network, and numerical model data from NOAA’s 13-km Rapid Refresh model. A regression is performed in the final layer of the network using NEXRAD-derived data mapped onto a 1-km grid as a target variable. The outputs of the CNN are fused with analyses from NEXRAD to create seamless radar mosaics that extend to offshore sectors and beyond. The model is calibrated and validated using both NEXRAD and spaceborne radar from NASA’s Global Precipitation Measurement (GPM) Mission’s Core Observatory satellite. The advantages over a random forest–based approach used in previous works are discussed.