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Book Cover
E-book
Author Čekanavičius, V

Title Compound Poisson Approximation
Published Milton : CRC Press LLC, 2024

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Description 1 online resource (320 p.)
Contents Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Contents -- Preface -- Authors -- Notation and abbreviations -- 1. Preliminaries -- 1.1. Notation -- 1.2. Basic properties of a compound Poisson distribution -- Exercises -- 2. Poisson limit theorem -- 2.1. Independent random variables -- 2.2. Dependent Bernoulli randomvariables -- 2.3. Proofs -- Exercises -- 3. Accuracy of Poisson approximation -- 3.1. Independent Bernoulli r.v.s -- 3.1.1. Estimates in terms of the total variation distance -- 3.1.2. Estimates in terms of the Gini-Kantorovich distance
3.7.1. Mixed Poisson distribution -- 3.7.2. Sum of 0-1 r.v.s till the stopping time -- 3.8. Proofs -- Exercises -- 4. Applications -- 4.1. Long head runs -- 4.2. Longmatch patterns -- 4.3. The Zubkov-Mikhailov statistic -- Exercises -- 5. Compound Poisson limit theorem -- 5.1. Compound Poisson limit theorem for independent r.v.s -- 5.2. Compound Poisson limit theorem for dependent r.v.s -- 5.3. Proofs -- Exercises -- 6. Accuracy of CP approximation: rare events -- 6.1. Independent random variables -- 6.2. Integer-valued random variables -- 6.3. Asymptotic expansions
6.4. Dependent random variables -- 6.5. Applications -- 6.5.1. m-run statistic -- 6.5.2. Long match patterns -- 6.5.3. Non-decreasing runs of fixed length -- 6.6. Proofs -- Exercises -- 7. Accuracy of CP approximation: discrete random variables -- 7.1. Independent Bernoulli random variables -- 7.2. Independent discrete random variables -- 7.3. Symmetric integer-valued random variables -- 7.4. Discrete non-lattice distributions -- 7.5. Applications -- 7.6. Proofs -- Exercises -- 8. Sums of m-dependent integer-valued random variables -- 8.1. Compound Poisson approximation -- 8.2. Applications
8.3. Proofs -- 8.3.1. Auxiliary results -- 8.3.2. Proofs of Theorems 8.1-8.8 -- Exercises -- 9. Markov Binomial distribution -- 9.1. Accuracy of approximation -- 9.2. Some other results -- 9.3. Proofs -- 9.3.1. Auxiliary results -- 9.3.2. Proofs of Theorems 9.1-9.6 -- Exercises -- 10. Compound Poisson approximations related to Kolmogorov's problem -- 10.1. Kolmogorov's first problem -- 10.2. Kolmogorov's second problem -- 10.3. Shifted compound Poisson approximation -- 10.4. Symmetric random variables -- 10.5. Random variables with non-negative ch.f.s -- 10.6. Proofs -- Exercises
Summary Compound Poisson Approximation appears naturally in situations where one deals with a large number of rare events. It has important applications in insurance, extreme value theory, reliability theory, mathematical biology, etc. This book synthesizes the most important recent research in the field in a single volume
Notes Description based upon print version of record
11. Compound Poisson approximation for sequences of random variables
Form Electronic book
Author Novak, S. Y
ISBN 9781040037270
1040037275