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Many probability books are written by mathematicians and have the built in bias that the reader is assumed to be a mathematician coming to the material for its beauty. This textbook is geared towards beginning graduate students from a variety of disciplines whose primary focus is not necessarily mathematics for its own sake. Instead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance, and engineering.

Inhaltsverzeichnis

1. Sets and Events

Abstract
The core classical theorems in probability and statistics are the following:
• The law of large numbers (LLN): Suppose {X n ,n ≥ 1} are independent, identically distributed (iid) random variables with common mean E(X n ) = μ. The LLN says the sample average is approximately equal to the mean, so that
$$\frac{1}{n}\sum\limits_{i = 1}^n {{X_i}} \to \mu$$
.
An immediate concern is what does the convergence arrow “→” mean? This result has far-reaching consequences since, if
$${X_i}\left\{ {\begin{array}{*{20}{c}} {1,}&{ifeventAoccurs,} \\ {0,}&{otherwise} \end{array}} \right.$$
then the average $$\sum\nolimits_{i = 1}^n {{X_i}} /n$$ is the relative frequency of occurrence of A in n repetitions of the experiment and μ = P(A). The LLN justifies the frequency interpretation of probabilities and much statistical estimation theory where it underlies the notion of consistency of an estimator.
• Central limit theorem (CLT): The central limit theorem assures us that sample averages when centered and scaled to have mean 0 and variance 1 have a distribution that is approximately normal. If {X n , n ≥ 1} are iid with common mean E(X n ) = μ and variance Var(X n ) = σ 2, then
$$P\left[ {\frac{{\sum\nolimits_{i = 1}^n {{X_i} - n\mu } }}{{\sigma \sqrt n }}x} \right] \to N(x): = \int_{ - \infty }^x {\frac{{{e^{ - {u^2}/2}}}}{{\sqrt {2\pi } }}} du$$
.
This result is arguably the most important and most frequently applied result of probability and statistics. How is this result and its variants proved?
• Martingale convergence theorems and optional stopping: A martingale is a stochastic process {X n , n ≥ 0} used to model a fair sequence of gambles (or, as we say today, investments). The conditional expectation of your wealth X n +1 after the next gamble or investment given the past equals the current wealth X n . The martingale results on convergence and optimal stopping underlie the modern theory of stochastic processes and are essential tools in application areas such as mathematical finance. What are the basic results and why do they have such far reaching applicability?
Sidney I. Resnick

2. Probability Spaces

Abstract
This chapter discusses the basic properties of probability spaces, and in particular, probability measures. It also introduces the important ideas of set induction.
Sidney I. Resnick

3. Random Variables, Elements, and Measurable Maps

Abstract
In this chapter, we will precisely define a random variable. A random variable is a real valued function with domain Ω which has an extra property called measurability that allows us to make probability statements about the random variables.
Sidney I. Resnick

4. Independence

Abstract
Independence is a basic property of events and random variables in a probability model. Its intuitive appeal stems from the easily envisioned property that the occurrence or non-occurrence of an event has no effect on our estimate of the probability that an independent event will or will not occur. Despite the intuitive appeal, it is important to recognize that independence is a technical concept with a technical definition which must be checked with respect to a specific probability model. There are examples of dependent events which intuition insists must be independent, and examples of events which intuition insists cannot be independent but still satisfy the definition. One really must check the technical definition to be sure.
Sidney I. Resnick

5. Integration and Expectation

Abstract
One of the more fundamental concepts of probability theory and mathematical statistics is the expectation of a random variable. The expectation represents a central value of the random variable and has a measure theory counterpart in the theory of integration.
Sidney I. Resnick

6. Convergence Concepts

Abstract
Much of classical probability theory and its applications to statistics concerns limit theorems; that is, the asymptotic behavior of a sequence of random variables. The sequence could consist of sample averages, cumulative sums, extremes, sample quantiles, sample correlations, and so on. Whereas probability theory discusses limit theorems, the theory of statistics is concerned with large sample properties of statistics, where a statistic is just a function of the sample.
Sidney I. Resnick

7. Laws of Large Numbers and Sums of Independent Random Variables

Abstract
This chapter deals with the behavior of sums of independent random variables and with averages of independent random variables. There are various results that say that averages of independent (and approximately independent) random variables are approximated by some population quantity such as the mean. Our goal is to understand these results in detail.
Sidney I. Resnick

8. Convergence in Distribution

Abstract
This chapter discusses the basic notions of convergence in distribution. Given a sequence of random variables, when do their distributions converge in a useful way to a limit?
Sidney I. Resnick

9. Characteristic Functions and the Central Limit Theorem

Abstract
This chapter develops a transform method calledcharacteristic functions for dealing with sums of independent random variables.
Sidney I. Resnick

10. Martingales

Abstract
Martingales are a class of stochastic processes which has had profound influence on the development of probability and stochastic processes. There are few areas of the subject untouched by martingales. We will survey the theory and applications of discrete time martingales and end with some recent developments in mathematical finance. Here is what to expect in this chapter:
• Absolute continuity and the Radon-Nikodym Theorem.
• Conditional expectation.
• Martingale definitions and elementary properties and examples.
• Martingale stopping theorems and applications.
• Martingale convergence theorems and applications.
• The fundamental theorems of mathematical finance.
Sidney I. Resnick

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