Introduction to Stochastic Processes - Lecture Notes (With 33 Illustrations)
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Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Contents 1 Probability review 4 1.1 Random variables . 4 1.2 Countable sets . 5 1.3 Discrete random variables . 5 1.4 Expectation . 7 1.5 Events and probability . 8 1.6 Dependence and independence . 9 1.7 Conditional probability . 10 1.8 Examples . 12 2 Mathematica in 15 min 15 2.1 Basic Syntax . 15 2.2 Numerical Approximation . 16 2.3 Expression Manipulation . 16 2.4 Lists and Functions . 17 2.5 Linear Algebra . 19 2.6 Predeﬁned Constants . 20 2.7 Calculus . 20 2.8 Solving Equations . 22 2.9 Graphics . 22 2.10 Probability Distributions and Simulation . 23 2.11 Help Commands . 24 2.12 Common Mistakes . 25 3 Stochastic Processes 26 3.1 The canonical probability space . 27 3.2 Constructing the Random Walk . 28 3.3 Simulation . 29 3.3.1 Random number generation . 29 3.3.2 Simulation of Random Variables . 30 3.4 Monte Carlo Integration . 33 4 The Simple Random Walk 35 4.1 Construction . 35 4.2 The maximum . 36 1 CONTENTS 5 Generating functions 40 5.1 Deﬁnition and ﬁrst properties . 40 5.2 Convolution and moments . 42 5.3 Random sums and Wald’s identity . 44 6 Random walks - advanced methods 48 6.1 Stopping times . 48 6.2 Wald’s identity II . 50 6.3 The distribution of the ﬁrst hitting time T1 .......................... 52 6.3.1 A recursive formula . 52 6.3.2 Generating-function approach . 53 6.3.3 Do we actually hit 1 sooner or later? . 55 6.3.4 Expected time until we hit 1? .............................. 55 7 Branching processes 56 7.1 A bit of history . 56 7.2 A mathematical model . 56 7.3 Construction and simulation of branching processes . 57 7.4 A generating-function approach . 58 7.5 Extinction probability . 61 8 Markov Chains 63 8.1 The Markov property . 63 8.2 Examples . 64 8.3 Chapman-Kolmogorov relations . 70 9 The “Stochastics” package 74 9.1 Installation . 74 9.2 Building Chains . 74 9.3 Getting information about a chain . 75 9.4 Simulation . 76 9.5 Plots . 76 9.6 Examples . 77 10 Classiﬁcation of States 79 10.1 The Communication Relation . 79 10.2 Classes . 81 10.3 Transience and recurrence . 83 10.4 Examples . 84 11 More on Transience and recurrence 86 11.1 A criterion for recurrence . 86 11.2 Class properties . 88 11.3 A canonical decomposition . 89 Last Updated: December 24, 2010 2 Intro to Stochastic Processes: Lecture Notes CONTENTS 12 Absorption and reward 92 12.1 Absorption . 92 12.2 Expected reward . 95 13 Stationary and Limiting Distributions 98 13.1 Stationary and limiting distributions . 98 13.2 Limiting distributions . 104 14 Solved Problems 107 14.1 Probability review . 107 14.2 Random Walks . 111 14.3 Generating functions . 114 14.4 Random walks - advanced methods . 120 14.5 Branching processes . 122 14.6 Markov chains - classiﬁcation of states . 133 14.7 Markov chains - absorption and reward . 142 14.8 Markov chains - stationary and limiting distributions . 148 14.9 Markov chains - various multiple-choice problems . 156 Last Updated: December 24, 2010 3 Intro to Stochastic Processes: Lecture Notes Chapter 1 Probability review The probable is what usually happens. —Aristotle It is a truth very certain that when it is not in our power to determine. what is true we ought to follow what is most probable —Descartes - “Discourse on Method” It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge. —Pierre Simon Laplace - “Théorie Analytique des Probabilités, 1812 ” Anyone who considers arithmetic methods of producing random digits is, of course, in a state of sin. —John von Neumann - quote in “Conic Sections” by D. MacHale I say unto you: a man must have chaos yet within him to be able to give birth to a dancing star: I say unto you: ye have chaos yet within you . —Friedrich Nietzsche - “Thus Spake Zarathustra” 1.1 Random variables Probability is about random variables. Instead of giving a precise deﬁnition, let us just metion that a random variable can be thought of as an uncertain, numerical (i.e., with values in R) quantity. While it is true that we do not know with certainty what value a random variable X will take, we usually know how to compute the probability that its value will be in some some subset of R. For example, we might be interested in P[X ≥ 7], P[X 2 [2; 3:1]] or P[X 2 f1; 2; 3g]. The collection of all such probabilities is called the distribution of X. One has to be very careful not to confuse the random variable itself and its distribution. This point is particularly important when several random variables appear at the same time. When two random variables X and Y have the same distribution, i.e., when P[X 2 A] = P[Y 2 A] for any set A, we say that X and Y are equally (d) distributed and write X = Y . 4 CHAPTER 1. PROBABILITY REVIEW 1.2 Countable sets Almost all random variables in this course will take only countably many values, so it is probably a good idea to review breiﬂy what the word countable means. As you might know, the countable inﬁnity is one of many diﬀerent inﬁnities we encounter in mathematics. Simply, a set is countable if it has the same number of elements as the set N = f1; 2;::: g of natural numbers. More precisely, we say that a set A is countable if.