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A random or stochastic process is a random variable that evolves in time by some random mechanism (of course, the time variable can be replaced by a space variable, or some other variable, in application). The variable can have a discrete set of values at a given time, or a continuum of values may be available. Likewise, the time variable can be discrete or continuous. A stochastic process is regarded as completely described if the probability distribution is known for all possible sets of times. A stationary process is one which has no absolute time origin. All probabilities are independent of a shift in the origin of time. This chapter discusses multitime probability description, conditional probabilities, stationary, Gaussian, and Markovian processes, and the Chapman–Kolmogorov condition.

This chapter examines a random process (X(t):t ≥ 0) taking values in R, that is governed by the events of an independent renewal process N(t), as follows: whenever an event of N(t) occurs, the process X(t) is restarted and runs independently of the past with initial value that has the same distribution as X(0). The case when each segment of the process between consecutive events of N(t) is a diffusion is studied, and expressions for the characteristic function of X(t) and its stationary distribution as t → ∞ are presented. An expression is derived for the expected first-passage time of X(t) to any value a, and several explicit examples of interest are considered. The chapter presents two approaches: first, it uses Wald's equation which supplies the mean in quite general circumstances; second, it explores possibilities for use of the moment-generating function of the first-passage time.

This chapter extends the ideas of Chapter 2 by introducing the notion of stochastic process or random process and its distribution functions. The idea of a Markov process is defined. The Chapman–Kolmagorov equation for Markov processes is given. From this, the Fokker–Planck equation for homogeneous continuous Markov processes is derived. The calculus of stochastic processes is then discussed, i.e., questions of what is meant by convergence, continuity, integration, Fourier analysis. The chapter concludes with a short discussion of white noise, a completely random process.

This chapter quantifies the dynamics of a crossword puzzle by using a network structure to model it. Specifically, the chapter determines how the interaction between the structure of cells in the puzzle and the difficulty of the clues affects the puzzle's solvability. It first builds an iterative stochastic process that exactly describes the solution and obtains its deterministic approximation, which gives a very simple fixed-point equation to solve for the final solution proportion. The chapter then shows via simulation on actual crosswords from the Sunday edition of The New York Times that certain network properties inherent to actual crossword networks are important predictors of the final solution size of the puzzle.

This chapter examines randomness and how it applies (or does not) in both abstract mathematics and in physical systems. There are many different ways in which a sequence of events could be said to be ‘random’. The mathematical theory of random processes, sometimes called stochastic processes, depends on being able to construct joint probabilities of large sequences of random variables, which can be very tricky to say the least. There are, however, some kinds of random processes where the theory is relatively straightforward. One class is when the sequence has no memory at all; this type of sequence is sometimes called white noise. Random processes can be either stationary or ergodic. The chapter also discusses predictability in principle and practice, and explains why pulling numbers out of an address book leads to a distribution of first digits that is not at all uniform. Aside from sequences of variables, other manifestations of randomness include points, patterns, and Poisson distribution.

Consider two physical problems describable by the same random process. The first process is the radioactive decay of a collection of nuclei. The second is the production of photoelectrons by a steady beam of light on a photodetector. In both cases, we can let a discrete, positive, integer valued, variable n(t) represent the number of counts emitted in the time interval between 0 and t. This is the essence of the Poisson process, an example of Markovian process. Other examples of Markovian processes include the one dimensional random walk, gambler's ruin, diffusion processes and the Einstein relation, Brownian motion, Langevin theory of velocities in Brownian motion, Langevin theory of positions in Brownian motion, and chaos.

Chapter VIII presents a novel mathematical theory of non-linear cointegration among second-order random processes. It begins by explaining why the accepted characterization of integrated second-order processes is inadequate for the analysis of non-linearly cointegrated economic systems, proposes alternative characterization of integrated processes, and develops novel ideas of non-linearly cointegrated second-order random processes. Thereafter, it presents a data confrontation of an economic theory about the dynamics of spot rates in foreign exchange whose variables share the behavior characteristics of the random processes in the mathematical theory. The data consist of weekly observations of a triple of exchange rates, and the empirical analysis is carried out in two ways – one by the methods of formal econometrics and another by the methods of present-day econometrics. The results of the two empirical analyses differ in interesting ways. Both agree that the behavior of the three exchange rates has the characteristics on which the mathematical theory insists, but their description of the dynamics of foreign exchange differ. Also, the present-day econometrics analysis rejects the empirical relevance of the given economic theory while the formal econometrics analysis accepts it. The acceptance of the theory carries interesting information about the dynamics of foreign exchange in social reality.

Benford's law arises naturally in a variety of stochastic settings, including products of independent random variables, mixtures of random samples from different distributions, and iterations of random maps. This chapter provides the concepts and tools to analyze significant digits and significands for these basic random processes. Benford's law also arises in many other important fields of stochastics, such as geometric Brownian motion, random matrices, and Bayesian models, and the chapter may serve as a preparation for specialized literature on these advanced topics. By Theorem 4.2 a random variable X is Benford if and only if log ¦X¦ is uniformly distributed modulo one.

This chapter embarks on a brief discussion of the mathematical theory of Benford's law. This law is the observation that in many collections of numbers, be they mathematical tables, real-life data, or combinations thereof, the leading significant digits are not uniformly distributed, as might be expected, but are heavily skewed toward the smaller digits. More specifically, Benford's law states that the significant digits in many data sets follow a very particular logarithmic distribution. The chapter lays out the basic theory of Benford's law before highlighting its more specific components: the significant digits and the significand (function), as well as the Benford property and its four characterizations. Finally, the chapter presents the basic theory of Benford's law in the context of deterministic and random processes.

In this chapter, the concept of probability is introduced. The rolling of a die is an example of a random process: the face that comes up is subject to chance. In probability, the goal is to quantify such a random process. That is, we want to assign a number to it. This chapter introduces some basic terms used in the study of probability; by the end of the chapter, the reader will be able to define the following terms: sample space, outcome, discrete outcome, event, probability, probability distribution, trial, empirical distribution, and Law of Large Numbers. Using an example, the chapter focuses on a single characteristic and introduces basic vocabulary associated with probability.