We consider spectrally positive Lévy processes with regularly varying Lévy measure and study conditional limit theorems that describe the way that various rare events occur. Specifically, we are interested in the asymptotic behaviour of the distribution of the path of the Lévy process (appropriately scaled) up to some fixed time, conditionally on the event that the process exceeds a (large) positive value at that time. Another rare event we study is the occurrence of a large maximum value up to a fixed time, and the corresponding asymptotic behaviour of the (scaled) Lévy process path. We study these distributional limit theorems both for a centred Lévy process and for one with negative drift. In the latter case, we also look at the reflected process, which is of importance in applications. Our techniques are based on the explicit representation of the Lévy process in terms of a two-dimensional Poisson random measure and merely use the Poissonian properties and regular variation estimates. We also provide a proof for the asymptotic behaviour of the tail of the stationary distribution for the reflected process. The work is motivated by earlier results for discrete-time random walks (e.g. Durrett (1980) and Asmussen (1996)) and also by their applications in risk and queueing theory.
Soil health can differ across cropping systems because of variation in edaphic and management factors. We evaluated how biological indicators of soil health (soil organic matter [SOM], permanganate oxidizable carbon [POXC], mineralizable carbon [MinC], autoclaved-citrate-extractable [ACE] protein, and potentially mineralizable nitrogen [PMN]) compared across four common Wisconsin cropping systems: grazed cool-season pastures, forage-based rotations that included perennial legumes or grasses, annual rotations receiving manure, and annual rotations receiving synthetic fertilizers. Biological indicators of soil health were up to 195% greater in pastures than other cropping systems. MinC, POXC and PMN were 10%-90% greater in forage-based rotations than annual cropping systems, but only MinC and POXC were greater in annual systems with manure compared to those without manure by 35% and 7%, respectively. Perennial vegetation and livestock integration offer the greatest potential to increase biological indicators of soil health in agricultural lands.
The accumulation of soil organic matter (SOM) is vital to the agronomic and environmental functioning of agroecosystems, yet the relative influence of inherent soil properties and agricultural management practices on SOM dynamics are not often addressed in individual studies. Using a network of 218 operating farm fields across Wisconsin and southern Minnesota, USA, this research employs single variable analysis (ANOVA and regression) and regression tree analysis to assess the effects of soil properties (texture, drainage class, pH) and management variables related to crop rotation, tillage, cover cropping, and manure application on SOM, as well as total organic carbon (TOC) and total nitrogen (TN) in the upper 15 cm. Single variable analysis revealed that greater SOM, TOC, and TN were associated with poorly drained soil, tile drained fields, high-clay content soil, and high biomass crop rotations. Soil organic matter (SOM) and TOC were strongly related (R2=0.71), but different regression trees were produced; SOM was most influenced by clay content, while TOC was most influenced by drainage class. Future benchmarking of SOM should be conducted with drainage and texture class categories. The rapid building of data sets thorough unstructured sampling, including an abundance of meta-data, should be a research priority in agricultural science.
We consider spectrally positive Lévy processes with regularly varying Lévy measure and study conditional limit theorems that describe the way that various rare events occur. Specifically, we are interested in the asymptotic behaviour of the distribution of the path of the Lévy process (appropriately scaled) up to some fixed time, conditionally on the event that the process exceeds a (large) positive value at that time. Another rare event we study is the occurrence of a large maximum value up to a fixed time, and the corresponding asymptotic behaviour of the (scaled) Lévy process path. We study these distributional limit theorems both for a centred Lévy process and for one with negative drift. In the latter case, we also look at the reflected process, which is of importance in applications. Our techniques are based on the explicit representation of the Lévy process in terms of a two-dimensional Poisson random measure and merely use the Poissonian properties and regular variation estimates. We also provide a proof for the asymptotic behaviour of the tail of the stationary distribution for the reflected process. The work is motivated by earlier results for discrete-time random walks (e.g. Durrett (1980) and Asmussen (1996)) and also by their applications in risk and queueing theory.
The accumulation of soil organic matter (SOM) is vital to the agronomic and environmental functioning of agroecosystems, yet the relative influence of inherent soil properties and agricultural management practices on SOM dynamics are not often addressed in individual studies. Using a network of 218 operating farm fields across Wisconsin and southern Minnesota, USA, this research employs single variable analysis (ANOVA and regression) and regression tree analysis to assess the effects of soil properties (texture, drainage class, and pH) and management variables related to crop rotation, tillage, cover cropping, and manure application on SOM, as well as total organic carbon (TOC) and total nitrogen (TN) in the upper 15 cm. Single variable analysis revealed that greater SOM, TOC, and TN were associated with poorly drained soil, tile‐drained fields, high clay content soil, and high biomass crop rotations. SOM and TOC were strongly related (R2 = 0.71), but different regression trees were produced; SOM was most influenced by clay content, while TOC was most influenced by drainage class. Future assessment for the building of SOM or TOC should be conducted with drainage and texture class categories and on a regional basis, given that these factors influence the practices that occur within landscapes. A rapid building of datasets through unstructured sampling, including an abundance of metadata, should be a research priority in agricultural science to identify practices to build SOM on a regional basis.
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