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This book offers a detailed review of perturbed random walks, perpetuities, and random processes with immigration. Being of major importance in modern probability theory, both theoretical and applied, these objects have been used to model various phenomena in the natural sciences as well as in insurance and finance. The book also presents the many significant results and efficient techniques and methods that have been worked out in the last decade. The first chapter is devoted to perturbed random walks and discusses their asymptotic behavior and various functionals pertaining to them, including supremum and first-passage time. The second chapter examines perpetuities, presenting results on continuity of their distributions and the existence of moments, as well as weak convergence of divergent perpetuities. Focusing on random processes with immigration, the third chapter investigates the existence of moments, describes long-time behavior and discusses limit theorems, both with and without scaling. Chapters four and five address branching random walks and the Bernoulli sieve, respectively, and their connection to the results of the previous chapters. With many motivating examples, this book appeals to both theoretical and applied probabilists.

Additive process (Probability theory) --- Random walk process (Mathematics) --- Walks, Random (Mathematics) --- Random walks (Mathematics) --- Stochastic processes --- Distribution (Probability theory. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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The collection of survey data from war zones or other unstable security situations is vulnerable to error because conflict often limits the implementation options. Although there are elevated risks throughout the process, this paper focuses specifically on challenges to frame construction and sample selection. The paper uses simulations based on data from the Mogadishu High Frequency Survey Pilot to examine the implications of the choice of second-stage selection methodology on bias and variance. Among the other findings, the simulations show the bias introduced by a random walk design leads to the underestimation of the poverty headcount by more than 10 percent. The paper also discusses the experience of the authors in the time required and technical complexity of the associated back-office preparation work and weight calculations for each method. Finally, as the simulations assume perfect implementation of the design, the paper also discusses practicality, including the ease of implementation and options for remote verification, and outlines areas for future research and pilot testing.

Administrative Records --- Age --- Algorithms --- Back Office --- Best Practice --- Business --- Calculation --- Case --- Cell Phones --- Classification --- Clustering --- Computer --- Confidence Intervals --- Counting --- Data --- Data Collection --- Description --- Document --- Effects --- Enumeration --- Equipment --- Errors --- Estimates --- Estimating --- Gps --- Human Error --- Image --- Implementation Plans --- Implementations --- Information --- Interviews --- Measurement --- Measures --- Methodology --- Methods --- Missing Values --- Modeling --- Monitoring --- Navigation --- Network --- Object --- Open Access --- Performance --- Phones --- Pilot Testing --- Precision --- Prediction --- Probability --- Probability Samples --- Protocol --- Random Sampling --- Random Walk --- Research --- Research Working Papers --- Researchers --- Result --- Risk --- Routing --- Sample Design --- Sample Size --- Samples --- Sampling --- Sampling Designs --- Satellite --- Scenarios --- Search --- Security --- Simulation --- Size --- Smart Phones --- Software --- Space --- Standard --- Standard Deviation --- Statistics --- Supervision --- Survey Data --- Survey Methodology --- Surveys --- Target --- Technical Training --- Techniques --- Technology --- Testing --- Theory --- Time --- URL --- Uses --- Variables --- Verification --- Web --- Weight --- Weighting --- WWW