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This text on stochastic processes and their applications is based on a set of lectures given during the past several years at the University of California, Santa Barbara (UCSB). It is an introductory graduate course designed for classroom purposes. Its objective is to provide graduate students of statistics with an overview of some basic methods and techniques in the theory of stochastic processes. The only prerequisites are some rudiments of measure and integration theory and an intermediate course in probability theory. There are more than 50 examples and applications and 243 problems and complements which appear at the end of each chapter. The book consists of 10 chapters. Basic concepts and definitions are pro­ vided in Chapter 1. This chapter also contains a number of motivating ex­ amples and applications illustrating the practical use of the concepts. The last five sections are devoted to topics such as separability, continuity, and measurability of random processes, which are discussed in some detail. The concept of a simple point process on R+ is introduced in Chapter 2. Using the coupling inequality and Le Cam's lemma, it is shown that if its counting function is stochastically continuous and has independent increments, the point process is Poisson. When the counting function is Markovian, the sequence of arrival times is also a Markov process. Some related topics such as independent thinning and marked point processes are also discussed. In the final section, an application of these results to flood modeling is presented.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Basic Concepts and Definitions

Abstract
Generally speaking, a stochastic or random process (in this book both terms will be used in an equivalent sense) is a family of random variables defined on a common probability space, indexed by the elements of an ordered set T, which is called the parameter set. Most often, T is taken to be an interval of time and the random variable indexed by an element tT is said to describe the state of the process at time t.
Petar Todorovic

Chapter 2. The Poisson Process and Its Ramifications

Abstract
We begin by describing in an informal fashion the subject matter of this chapter. The part of the general theory of stochastic processes dealing with countable sets of points randomly distributed on the real line or in an arbitrary space (for instance, Cartesian d-dimensional space) is called the “Theory of Point Processes.” Of all point processes, those on the real line have been most widely studied. Notwithstanding their relatively simple structure, they form building blocks in a variety of industrial, biological, geophysical, and engineering applications. The following example describes a general situation which in a natural fashion introduces a point process on a line.
Petar Todorovic

Chapter 3. Elements of Brownian Motion

Abstract
In Chapter 1 (Section 1.6, Example 1.3), we discussed in some detail the nature and causes of the random motion of a small colloidal-size particle submerged in water. According to kinetic theory, this movement is due to the thermal diffusion of the water molecules, which are incessantly bombarding the particle, forcing it to move constantly in a zigzag path. The phenomenon was named “Brownian motion” after R. Brown, an English botanist who was first to observe it. In 1904, H. Poincaré explained that large particles submerged in water do not move, notwithstanding a huge number of impacts from all directions by the molecules of the surrounding medium, simply because, according to the theory of large numbers, they neutralize each other.
Petar Todorovic

Chapter 4. Gaussian Processes

Abstract
In this section we present a review of some basic properties of the square matrices that will be needed throughout this chapter. It is expected that those who read this section have some background in matrix analysis.
Petar Todorovic

Chapter 5. L 2 Space

Abstract
In many applications of the theory of stochastic processes, an important role is played by families of square integrable (second-order) r.v.’s. In this section, we give some basic definitions and prove some fundamental inequalities involving second-order complex-valued r.v.’s.
Petar Todorovic

Chapter 6. Second-Order Processes

Abstract
There exists a large class of engineering and physics problems whose solutions require only the knowledge of the first two moments and some very general properties of a second-order random process (see Definition 1.5.8). This chapter is concerned with some key properties of complex-valued second-order random processes.
Petar Todorovic

Chapter 7. Spectral Analysis of Stationary Processes

Abstract
Let {ξ(t);tR} be a wide sense stationary, complex-valued random process with E{ξ(t)} = 0 and
$$ C\left( t \right) = E\left\{ {\xi \left( s \right)\overline {\xi \left( {s + t} \right)} } \right\}. $$
(7.1.1)
Petar Todorovic

Chapter 8. Markov Processes I

Abstract
The concept of a Markov random process was defined in Section 1.7 of Chapter 1. Various particular cases were discussed in subsequent chapters. Without doubt, this has been the most extensively studied class of random processes.
Petar Todorovic

Chapter 9. Markov Processes II: Application of Semigroup Theory

Abstract
Let {ξ(t);t ≥ 0} be a real homogeneous Markov process with transition probability P(x, t, B). In applications the following situation is typical. The transition probability is known for all t in a neighborhood of the origin. Then, P(x, t, B) can be determined for allt > 0 by means of the Chapman- Kolmogorov equation (8.1.2).
Petar Todorovic

Chapter 10. Discrete Parameter Martingales

Abstract
The concept of a martingale introduced in Section 1.5 of Chapter 1 [see (1.5.19)] was defined in terms of the conditional expectation with respect to a σ-algebra. In this section, we will explore briefly some basic properties of this conditional expectation, which are needed in this chapter. We begin with some definitions.
Petar Todorovic

Backmatter

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