Ntroduction to Stochastic Processes Erhan Cinlar

Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. In a lively and imaginative presentation, studded with examples, exercises, and applications, and supported by inclusion of computational procedures, the author has created a textbook that provides easy access to this fundamental topic for many students of applied sciences at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is accessible to beginners but flexible enough to serve as well those who come to the course with strong backgrounds.

This book has been designed for a final year undergraduate course in stochastic processes. It will also be suitable for mathematics undergraduates and others with interest in probability and stochastic processes, who wish to study on their own. The main prerequisite is probability theory: probability measures, random variables, expectation, independence, conditional probability, and the laws of large numbers. The only other prerequisite is calculus.

The volume Stochastic Processes by K. Itö was published as No. 16 of Lecture Notes Series from Mathematics Institute, Aarhus University in August, 1969, based on Lectures given at that Institute during the academie year 1968­ 1969. The volume was as thick as 3.5 cm., mimeographed from typewritten manuscript and has been out of print for many years. Since its appearance, it has served, for those abIe to obtain one of the relatively few copies available, as a highly readable introduetion to basic parts of the theories of additive processes (processes with independent increments) and of Markov processes. It contains, in particular, a clear and detailed exposition of the Lévy-It ö decomposition of additive processes. Encouraged by Professor It ó we have edited the volume in the present book form, amending the text in a number of places and attaching many footnotes. We have also prepared an index. Chapter 0 is for preliminaries. Here centralized sums of independent ran­ dom variables are treated using the dispersion as a main tooI. Lévy's form of characteristic functions of infinitely divisible distributions and basic proper­ ties of martingales are given. Chapter 1 is analysis of additive processes. A fundamental structure the­ orem describes the decomposition of sample functions of additive processes, known today as the Lévy-Itó decomposition. This is thoroughly treated, as­ suming no continuity property in time, in a form close to the original 1942 paper of Itó, which gave rigorous expression to Lévy's intuitive understanding of path behavior.

The present reprint contains all of the articles accepted and published in the Special Issue "Probability and Stochastic Processes with Applications to Communications, Systems and Networks" from the MDPI Mathematics journal. This Special Issue is devoted to probability, statistics, stochastic processes and their different applications in system and network analysis. The Special Issue includes works related to the analysis and applications of different queuing models, which begin with general approaches in modeling queuing systems and networks; an analysis of probabilistic and statistical methods in telecommunication; an asymptotic analysis of queuing networks in the condition of a large load; general complex networks and their structures, e.g., topology and graph theory; mathematical methods and models in smart cities; exclusive statistical methods, such as statistical estimates in bio/ecology, medicine and neural networks; and studies that estimate parameters in complex technical systems, etc. We hope that the scientific results collected in this reprint will help foster future research related to probability, stochastic processes and their applications.

Applied Probability and Stochastic Processes is an edited work written in honor of Julien Keilson. This volume has attracted a host of scholars in applied probability, who have made major contributions to the field, and have written survey and state-of-the-art papers on a variety of applied probability topics, including, but not limited to: perturbation method, time reversible Markov chains, Poisson processes, Brownian techniques, Bayesian probability, optimal quality control, Markov decision processes, random matrices, queueing theory and a variety of applications of stochastic processes. The book has a mixture of theoretical, algorithmic, and application chapters providing examples of the cutting-edge work that Professor Keilson has done or influenced over the course of his highly-productive and energetic career in applied probability and stochastic processes. The book will be of interest to academic researchers, students, and industrial practitioners who seek to use the mathematics of applied probability in solving problems in modern society.