A splash experiment was carried out on a model soil–glass beads with a diameter of 425–600 μm using high‐speed cameras and sticky paper. Two different types of particles were involved in the process: droplets of water and glass beads. We argue that the result of splash of solid particles is best modeled as a stochastic point process, that is, a random number of randomly distributed points (beads) on a plane, and provide basic physical and statistical evidence that, in medium distance range (i.e., for our experiment, in the ranges of 29–64 mm), the splash may be modeled as the Poisson point process. We also argue that, in the range between 15 and 29 mm, a distribution different than Poisson is closer to reality. These two radically different types of distributions of numbers of beads in two regions reflect the fact that the solid phase of the splash involves two types of beads: those ejected in the early stage, traveling larger distances, and those ejected later, traveling shorter distances. Information on the distributions of and relations between the numbers of splashed particles in different regions may be instrumental in understanding mechanics and scale of the spread of pollutants/pathogens and plant diseases as a result of splash. Meanwhile, we describe the distributions of the total number of beads, the maximum range, and the average distance beads particles travel in a single experiment and discuss effectiveness of detection of beads by the cameras.
Keywords:Complete convergence Negatively associated random fields ρ * -mixing random fields Martingale random fields Baum-Katz theorem Let {X n , n ∈ N r } be a random field i.e. a family of random variables indexed by N r , r 2.We discuss complete convergence and convergence rates under assumption on dependence structure of random fields in the case of nonidentical distributions. Results are obtained for negatively associated random fields, ρ * -mixing random fields (having maximal coefficient of correlation strictly smaller then 1) and martingale random fields.
Let {X n , n ∈ N d } be a random field i.e. a family of randomComplete convergence, convergence rates for non identically distributed, negatively dependent and martingale random fields are studied by application of Fuk-Nagaev inequality. The results are proved in asymmetric convergence case i.e. for the norming sequence equal n α 1
We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for𝔹-valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.
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