Random Summation: Limit Theorems and

This book provides an introduction to the asymptotic theory of random summation. It describes solving problems in hardware and software reliability, engineering, finance, insurance, computer science, physics and biology. The authors focus on transfer theorems, the classes of limit laws, and convergence of distributions of sums for a number of random variables. Special sections deal with doubling with repair, rarefaction of renewal processes, limit theorems for supercritical Galton-Watson processes, information properties of probability distributions, and asymptotic behavior of doubly stochastic Poisson processes.

Examples
Examples Related to Generalized Poisson Laws
A Remarkable Formula of Queueing Theory
Other Examples
Doubling with Repair
Mathematical Model
A Limit Theorem for the Trouble-Free Performance Duration
The Class of Limit Laws
Some Properties of Limit Distributions
Domains of Geometrical Attraction of the Laws from Class c
Limit Theorems for "Growing" Random Sums
A Transfer Theorem. Limit Laws
Necessary and Sufficient Conditions for Convergence
Convergence to Distributions from Identifiable Families
Limit Theorems for Risk Processes
Some Models of Financial Mathematics
Rarefied Renewal Processes
Limit Theorems for Random Sums in the Double Array Scheme
Transfer Theorems. Limit Laws
Converses of the Transfer Theorems
Necessary and Sufficient Conditions for the Convergence of Random Sums of Independent Identically Distributed Random Variables
More on Some Models of Financial Mathematics
Limit Theorems for Supercritical Galton-Watson Processes
Randomly Infinitely Divisible Distributions
Mathematical Theory of Reliability Growth. A Bayesian Approach
Bayesian Reliability Growth Models
Conditionally Geometrical Models
Conditionally Exponential Models
Renewing Models
Models with Independent Decrements of Volumes of Defective Sets
Order-Statistics-Type (Mosaic) Reliability Growth Models
Generalized Conditionally Exponential Models
Statistical Prediction of Reliability by Renewing Models
Statistical Prediction of Reliability by Order-Statistics-Type Models
Appendix 1: Information Properties of Probability Distributions
Mathematical Models of Information and Uncertainty
Limit Theorems of Probability Theory and the Universal Principle of Non-Decrease of Uncertainty
Appendix 2: Asymptotic Behavior of Generalized Doubly Stochastic Poisson Processes
General Information on Doubly Stochastic Poisson Processes
A General Limit Theorem for Superpositions of Random Processes
Limit Theorem for Cox Processes
Limit Theorems for Generalized Cox Processes
Convergence Rate Estimates in Limit Theorems for Generalized Cox Processes
Asymptotic Expansions for Generalized Cox Processes
Estimates for the Concentration Functions of Generalized Cox Processes
Bibliographical Commentary
Index
References

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Random Summation: Limit Theorems and Applications

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Supplemental Materials