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This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such as Stochastic Networks, Analytic Combinatorics, and Quantum Physics. This second edition consists of two parts. Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By using Complex Function Theory, Boundary Value Problems, Riemann Surfaces, and Galois Theory, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize the Transient Behavior of the walks, as well as to find explicit solutions to the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable Systems. Part II borrows special case-studies from queueing theory (in particular, the famous problem of Joining the Shorter of Two Queues) and enumerative combinatorics (Counting, Asymptotics). Researchers and graduate students should find this book very useful.
Random walks (Mathematics) --- Additive process (Probability theory) --- Random walk process (Mathematics) --- Walks, Random (Mathematics) --- Mathematics. --- Mathematical statistics. --- Difference equations. --- Functional equations. --- Probabilities. --- Statistics. --- Probability Theory and Stochastic Processes. --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Probability and Statistics in Computer Science. --- Difference and Functional Equations. --- Stochastic processes --- Distribution (Probability theory. --- Computer science. --- Informatics --- Science --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Mathematics --- Econometrics --- Equations, Functional --- Functional analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistics . --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Statistical inference --- Statistics, Mathematical --- Statistics --- Sampling (Statistics) --- Probability --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Computer science --- Probability Theory. --- Statistics in Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Computer mathematics --- Electronic data processing
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This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such as Stochastic Networks, Analytic Combinatorics, and Quantum Physics. This second edition consists of two parts. Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By using Complex Function Theory, Boundary Value Problems, Riemann Surfaces, and Galois Theory, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize the Transient Behavior of the walks, as well as to find explicit solutions to the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable Systems. Part II borrows special case-studies from queueing theory (in particular, the famous problem of Joining the Shorter of Two Queues) and enumerative combinatorics (Counting, Asymptotics). Researchers and graduate students should find this book very useful.
Statistical science --- Functional analysis --- Differential equations --- Operational research. Game theory --- Probability theory --- Mathematics --- Computer science --- waarschijnlijkheidstheorie --- stochastische analyse --- computers --- mathematische modellen --- statistiek --- informatietechnologie --- wiskunde --- kansrekening --- computerkunde --- statistisch onderzoek
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