xvi Preface
to guide the reader in both the mathematical and intuitive understanding necessary in
developing and using stochastic process models in studying application areas.
Application-oriented students often ask why it is important to understand axioms,
theorems, and proofs in mathematical models when the precise results in the model
become approximations in the real-world system being modeled. One answer is that
a deeper understanding of the mathematics leads to the required intuition for under-
standing the differences between model and reality. Another answer is that theorems are
transferable between applications, and thus enable insights from one application area to
be transferred to another.
Given the need for precision in the theory, however, why is an axiomatic approach
needed? Engineering and science students learn to use calculus, linear algebra, and
undergraduate probability effectively without axioms or rigor. Why does this not work
for more advanced probability and stochastic processes?
Probability theory has more than its share of apparent paradoxes, and these show
up in very elementary arguments. Undergraduates are content with this, since they
can postpone these questions to later study. For the more complex issues in graduate
work, however, reasoning without a foundation becomes increasingly frustrating, and
the axioms provide the foundation needed for sound reasoning without paradoxes.
I have tried to avoid the concise and formal proofs of pure mathematics, and instead
use explanations that are longer but more intuitive while still being precise. This is partly
to help students with limited exposure to pure mathematics, and partly because intuition
is vital when going back and forth between a mathematical model and a real-world
problem. In doing research, we grope toward results, and successful groping requires
both a strong intuition and precise reasoning.
The text neither uses nor develops measure theory. Measure theory is undoubtedly
important in understanding probability at a deep level, but most of the topics useful in
many applications can be understood without measure theory. I believe that the level of
precision here provides a good background for a later study of measure theory.
The text does require some background in probability at an undergraduate level.
Chapter 1 presents this background material as a review, but it is too concentrated and
deep for most students without prior background. Some exposure to linear algebra and
analysis (especially concrete topics like vectors, matrices, and limits) is helpful, but the
text develops the necessary results. The most important prerequisite is the mathematical
maturity and patience to couple precise reasoning with intuition.
The organization of the text, after the review in Chapter 1 is as follows: Chapters 2,
3, and 4 treat three of the simplest and most important classes of stochastic processes,
first Poisson processes, next Gaussian processes, and finally finite-state Markov chains.
These are beautiful processes where almost everything is known, and they contribute
insights, examples, and initial approaches for almost all other processes. Chapter 5 then
treats renewal processes, which generalize Poisson processes and provide the foundation
for the rest of the text.
Chapters 6 and 7 use renewal theory to generalize Markov chains to countable state
spaces and continuous time. Chapters 8 and 10 then study decision making and estima-
tion, which in a sense gets us out of the world of theory and back to using the theory.
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