Probability Theory

Frontmatter

1. Basic Measure Theory

In this chapter, we introduce the classes of sets that allow for a systematic treatment of events and random observations in the framework of probability theory. Furthermore, we construct measures, in particular probability measures, on such classes of sets. Finally, we define random variables as measurable maps.

2. Independence

The measure theory from the preceding chapter is a linear theory that could not describe the dependence structure of events or random variables. We enter the realm of probability theory exactly at this point, where we define independence of events and random variables. Independence is a pivotal notion of probability theory, and the computation of dependencies is one of the theory’s major tasks.

3. Generating Functions

It is a fundamental principle of mathematics to map a class of objects that are of interest into a class of objects where computations are easier. This map can be one to one, as with linear maps and matrices, or it may map only some properties uniquely, as with matrices and determinants.

4. The Integral

Based on the notions of measure spaces and measurable maps, we introduce the integral of a measurable map with respect to a general measure. This generalises the Lebesgue integral that can be found in textbooks on calculus. Furthermore, the integral is a cornerstone in a systematic theory of probability that allows for the definition and investigation of expected values and higher moments of random variables.

5. Moments and Laws of Large Numbers

The most important characteristic quantities of random variables are the median, expectation and variance. For large n, the expectation describes the typical approximate value of the arithmetic mean (X₁ + … + X _n )/n of i.i.d. random variables (law of large numbers). In Chapter 15, we will see how the variance determines the size of the typical deviations of the arithmetic mean from the expectation.

6. Convergence Theorems

In the strong and the weak laws of large numbers, we implicitly introduced the notions of almost sure convergence and convergence in probability of random variables. We saw that almost sure convergence implies convergence in measure/probability. This chapter is devoted to a systematic treatment of almost sure convergence, convergence in measure and convergence of integrals. The key role for connecting convergence in measure and convergence of integrals is played by the concept of uniform integrability.

7. L p -Spaces and the Radon-Nikodym Theorem

In this chapter, we study the spaces of functions whose pth power is integrable. In Section 7.2, we first derive some of the important inequalities (Hölder, Minkowski, Jensen) and then in Section 7.3 investigate the case p = 2 in more detail. Apart from the inequalities, the important results for probability theory are Lebesgue’s decomposition theorem and the Radon-Nikodym theorem in Section 7.4. At first reading, some readers might wish to skip some of the more analytic parts of this chapter.

8. Conditional Expectations

If there is partial information on the outcome of a random experiment, the probabilities for the possible events may change. The concept of conditional probabilities and conditional expectations formalises the corresponding calculus.

9. Martingales

One of the most important concepts of modern probability theory is the martingale, which formalises the notion of a fair game. In this chapter, we first lay the foundations for the treatment of general stochastic processes. We then introduce martingales and the discrete stochastic integral. We close with an application to a model from mathematical finance.

10. Optional Sampling Theorems

In Chapter 9 we saw that martingales are transformed into martingales if we apply certain admissible gambling strategies. In this chapter, we establish a similar stability property for martingales that are stopped at a random time. In order also to obtain these results for submartingales and supermartingales, in the first section, we start with a decomposition theorem for adapted processes. We show the optional sampling and optional stopping theorems in the second section. The chapter finishes with the investigation of random stopping times with an infinite time horizon.

11. Martingale Convergence Theorems and Their Applications

We became familiar with martingales X = (X _n )_n∈N0 as fair games and found that under certain transformations (optional stopping, discrete stochastic integral) martingales turn into martingales. In this chapter, we will see that under weak conditions (non-negativity or uniform integrability) martingales converge almost surely. Furthermore, the martingale structure implies L ^p -convergence under assumptions that are (formally) weaker than those of Chapter 7. The basic ideas of this chapter are Doob’s inequality (Theorem 11.2) and the upcrossing inequality (Lemma 11.3).

12. Backwards Martingales and Exchangeability

With many data acquisitions, such as telephone surveys, the order in which the data come does not matter. Mathematically, we say that a family of random variables is exchangeable if the joint distribution does not change under finite permutations. De Finetti’s structural theorem says that an infinite family of E-valued exchangeable random variables can be described by a two-stage experiment. At the first stage, a probability distribution Ξ on E is drawn at random. At the second stage, i.i.d. random variables with distribution Ξ are implemented.

13. Convergence of Measures

One focus of probability theory is distributions that are the result of an interplay of a large number of random impacts. Often a useful approximation can be obtained by taking a limit of such distributions, for example, a limit where the number of impacts goes to infinity. With the Poisson distribution, we have encountered such a limit distribution that occurs as the number of very rare events when the number of possibilities goes to infinity (see Theorem 3.7). In many cases, it is necessary to rescale the original distributions in order to capture the behaviour of the essential fluctuations, e.g., in the central limit theorem. While these theorems work with real random variables, we will also see limit theorems where the random variables take values in more general spaces such as, for example, the space of continuous functions when we model the path of the random motion of a particle.

14. Probability Measures on Product Spaces

As a motivation, consider the following example. Let X be a random variable that is uniformly distributed on [0, 1]. As soon as we know the value of X, we toss n times a coin that has probability X for a success. Denote the results by Y₁,…, Y _n .

15. Characteristic Functions and the Central Limit Theorem

The main goal of this chapter is the central limit theorem (CLT) for sums of independent random variables (Theorem 15.37) and for independent arrays of random variables (Lindeberg-Feller theorem, Theorem 15.43). For the latter, we prove only that one of the two implications (Lindeberg’s theorem) that is of interest in the applications.

16. Infinitely Divisible Distributions

17. Markov Chains

In spite of their simplicity, Markov processes with countable state space (and discrete time) are interesting mathematical objects with which a variety of real-world phenomena can be modelled. We give an introduction to the basic concepts and then study certain examples in more detail. The connection with discrete potential theory will be investigated later, in Chapter 19. Some readers might prefer to skip the somewhat technical construction of general Markov processes in Section 17.1.

18. Convergence of Markov Chains

We consider a Markov chain X with invariant distribution π and investigate conditions under which the distribution of X _n converges to π for n → ∞. Essentially it is necessary and sufficient that the state space of the chain cannot be decomposed into subspaces

• that the chain does not leave
• or that are visited by the chain periodically; e.g., only for odd n or only for even n.

19. Markov Chains and Electrical Networks

20. Ergodic Theory

21. Brownian Motion

In Example 14.45, we constructed a (canonical) process (X _t )_t∈[0,∞) with independent stationary normally distributed increments. For example, such a process can be used to describe the motion of a particle immersed in water or the change of prices in the stock market. We are now interested in properties of this process X that cannot be described in terms of finite-dimensional distributions but reflect the whole path t → X _t . For example, we want to compute the distribution of the functional F(X) := sup_t∈[0,1] X _t . The first problem that has to be resolved is to show that F is a random variable.

In this chapter, we investigate continuity properties of paths of stochastic processes and show how they ensure measurability of some path functionals. Then we construct a version of X that has continuous paths, the so-called Wiener process or Brownian motion. Without exaggeration, it can be stated that Brownian motion is the central object of probability theory.

22. Law of the Iterated Logarithm

For sums of independent random variables we already know two limit theorems: the law of large numbers and the central limit theorem. The law of large numbers describes for large n ∈ ℕ, the typical behaviour, or average value behaviour, of sums of n random variables. On the other hand, the central limit theorem quantifies the typical fluctuations about this average value.

23. Large Deviations

24. The Poisson Point Process

Poisson point processes can be used as a cornerstone in the construction of very different stochastic objects such as, for example, infinitely divisible distributions, Markov processes with complex dynamics, objects of stochastic geometry and so forth.

25. The Itô Integral

The Itô integral allows us to integrate stochastic processes with respect to the increments of a Brownian motion or a somewhat more general stochastic process. We develop the Itô integral first for Brownian motion and then for generalised diffusion processes. In the third section, we derive the celebrated Itô formula. This is the chain rule for the Itô integral that enables us to do explicit calculations with the Itô integral. In the fourth section, we use the Itô formula to obtain a stochastic solution of the classical Dirichlet problem. This in turn is used in the fifth section in order to show that like symmetric simple random walk, Brownian motion is recurrent in low dimensions and transient in high dimensions.

26. Stochastic Differential Equations

Stochastic differential equations describe the time evolution of certain continuous Markov processes with values in ℝ ⁿ . In contrast with classical differential equations, in addition to the derivative of the function, there is a term that describes the random fluctuations that are coded as an Itô integral with respect to a Brownian motion. Depending on how seriously we take the concrete Brownian motion as the driving force of the noise, we speak of strong and weak solutions. In the first section, we develop the theory of strong solutions under Lipschitz conditions for the coefficients. In the second section, we develop the so-called (local) martingale problem as a method of establishing weak solutions. In the third section, we present some examples in which the method of duality can be used to prove weak uniqueness.

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