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Anthony Louis Almudevar, Edilson Fernandes de Arruda
Taylor & Francis Ltd
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Table Of Contents
1 Introduction PART I Mathematical background 2 Real analysis and linear algebra 2.1 Definitions and notation 2.1.1 Numbers, sets and vectors 2.1.2 Logical notation 2.1.3 Set algebra 2.1.4 The supremum and infimum 2.1.5 Rounding off 2.1.6 Functions 2.1.7 Sequences and limits 2.1.8 Infinite series 2.1.9 Geometric series 2.1.10 Classes of real valued functions 2.1.11 Graphs 2.1.12 The binomial coefficient 2.1.13 Stirling's approximation of the factorial 2.1.14 L'Hopital's rule 2.1.15 Taylor's theorem 2.1.16 The lp norm 2.1.17 Power means 2.2 Equivalence relationships 2.3 Linear algebra 2.3.1 Matrices 2.3.2 Eigenvalues and spectral decomposition 2.3.3 Symmetric, Hermitian and positive definite matrices 2.3.4 Positive matrices 2.3.5 Stochastic matrices 2.3.6 Nonnegative matrices and graph structure 3 Background - measure theory 3.1 Topological spaces 3.1.1 Bases of topologies 3.1.2 Metric space topologies 3.2 Measure spaces 3.2.1 Formal construction of measures 3.2.2 Completion of measures 3.2.3 Outer measure 3.2.4 Extension of measures 3.2.5 Counting measure 3.2.6 Lebesgue measure 3.2.7 Borel sets 3.2.8 Dynkin system theorem 3.2.9 Signed measures 3.2.10 Decomposition of measures 3.2.11 Measurable functions 3.3 Integration 3.3.1 Convergence of integrals 3.3.2 Lp spaces 3.3.3 Radon-Nikodym derivative 3.4 Product spaces 3.4.1 Product topologies 3.4.2 Product measures 3.4.3 The Kolmogorov extension theorem 4 Background - probability theory 4.1 Probability measures - basic properties 4.2 Moment generating functions (MGF) and cumulant generating functions (CGF) 4.2.1 Moments and cumulants 4.2.2 MGF and CGF of independent sums 4.2.3 Relationship of the CGF to the normal distribution 4.2.4 Probability generating functions 4.3 Conditional distributions 4.4 Martingales 4.4.1 Stopping times 4.5 Some important theorems 4.6 Inequalities for tail probabilities 4.6.1 Chernoff bounds 4.6.2 Chernoff bound for the normal distribution 4.6.3 Chernoff bound for the gamma distribution 4.6.4 Sample means 4.6.5 Some inequalities for bounded random variables 4.7 Stochastic ordering 4.7.1 MGF ordering of the gamma and exponential distribution 4.7.2 Improved bounds based on hazard functions 4.8 Theory of stochastic limits 4.8.1 Covergence of random variables 4.8.2 Convergence of measures 4.8.3 Total variation norm 4.9 Stochastic kernels 4.9.1 Measurability of measure kernels 4.9.2 Continuity of measure kernels 4.10 Convergence of sums 4.11 The law of large numbers 4.12 Extreme value theory 4.13 Maximum likelihood estimation 4.14 Nonparametric estimates of distributions 4.15 Total variation distance for discrete distributions 5 Background - stochastic processes 5.1 Counting processes 5.1.1 Renewal processes 5.1.2 Poisson process 5.2 Markov processes 5.2.1 Discrete state spaces 5.2.2 Global properties of Markov chains 5.2.3 General state spaces 5.2.4 Geometric ergodicity 5.2.5 Spectral properties of Markov chains 5.3 Continuous-time Markov chains 5.3.1 Birth and death processes 5.4 Queueing systems 5.4.1 Queueing systems as birth and death processes 5.4.2 Utilization factor 5.4.3 General queueing systems and embedded Markov chains 5.5 Adapted counting processes 5.5.1 Asymptotic behavior 5.5.2 Relationship to adapted events 6 Functional analysis 6.1 Metric spaces 6.1.1 Contractive mappings 6.2 The Banach fixed point theorem 6.2.1 Stopping rules for fixed point algorithms 6.3 Vector spaces 6.3.1 Quotient spaces 6.3.2 Basis of a vector space 6.3.3 Operators 6.4 Banach spaces 6.4.1 Banach spaces and completeness 6.4.2 Linear operators 6.5 Norms and norm equivalence 6.5.1 Norm dominance 6.5.2 Equivalence properties of norm equivalence classes 6.6 Quotient spaces and seminorms 6.7 Hilbert spaces 6.8 Examples of Banach spaces 6.8.1 Finite dimensional spaces 6.8.2 Matrix norms and the submultiplicative property 6.8.3 Weighted n
Dr. Almudevar was born in Halifax and raised in Ontario, Canada. He completed a PhD in Statistics at the University of Toronto, and is currently a faculty member in the Department of Biostatistics and Computational Biology at the University of Rochester. He has a wide range of interests, which include biological network modeling, analysis of genetic data, immunological modeling and clinical applications of technological home monitoring. He has a more general interest in optimization and control theory, with an emphasis on the computational issues associated with Markov decision processes.