Inversions of Levy Measures and the Relation Between Long and Short Time Behavior of Levy Processes

The inversion of a Levy measure was first introduced (under a different name) in Sato 2007. We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Levy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Levy process as time approaches infinity. This allows us to get self contained conditions for a Levy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Levy processes and for the important class of tempered stable Levy processes. For this latter class, we give detailed results in terms of their Rosinski measures

Mapping soil transmitted helminths and schistosomiasis under uncertainty: a systematic review and critical appraisal of evidence

Spatial modelling of STH and schistosomiasis epidemiology is now commonplace. Spatial epidemiological studies help inform decisions regarding the number of people at risk as well as the geographic areas that need to be targeted with mass drug administration; however, limited attention has been given to propagated uncertainties, their interpretation, and consequences for the mapped values. Using currently published literature on the spatial epidemiology of helminth infections we identified: (1) the main uncertainty sources, their definition and quantification and (2) how uncertainty is informative for STH programme managers and scientists working in this domain.We performed a systematic literature search using the Preferred Reporting Items for Systematic reviews and Meta-Analysis (PRISMA) protocol. We searched Web of Knowledge and PubMed using a combination of uncertainty, geographic and disease terms. A total of 73 papers fulfilled the inclusion criteria for the systematic review. Only 9% of the studies did not address any element of uncertainty, while 91% of studies quantified uncertainty in the predicted morbidity indicators and 23% of studies mapped it. In addition, 57% of the studies quantified uncertainty in the regression coefficients but only 7% incorporated it in the regression response variable (morbidity indicator). Fifty percent of the studies discussed uncertainty in the covariates but did not quantify it. Uncertainty was mostly defined as precision, and quantified using credible intervals by means of Bayesian approaches.None of the studies considered adequately all sources of uncertainties. We highlighted the need for uncertainty in the morbidity indicator and predictor variable to be incorporated into the modelling framework. Study design and spatial support require further attention and uncertainty associated with Earth observation data should be quantified. Finally, more attention should be given to mapping and interpreting uncertainty, since they are relevant to inform decisions regarding the number of people at risk as well as the geographic areas that need to be targeted with mass drug administration