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Introduction to Statistical Limit Theory

Alan M. Polansky
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Texts in Statistical Science
[Reviewed by
Peter Rabinovitch
, on

Introduction to Statistical Limit Theory covers the basic limit theorems of probability and their applications to statistics. It does not presuppose any measure theory, although a decent background in analysis is required (say at the level of Spivak's Calculus), and of course enough background in statistics to make the applications meaningful.

This book is clearly aimed at the student learning the material. The prose is clear and graphs are used well to illustrate concepts. Students will really like the large number of worked concrete examples. For example, many probability texts will show that the sum of two uniformly integrable random variables is uniformly integrable. This book will also show examples of several particular families that are uniformly integrable, such as random variables with a Gaussian distribution with mean 0 and variance v(n), 0 < v(n) < ∞.

Each chapter has exercises and R projects that would be very helpful to solidify the understanding of the material in the chapter. These and the worked examples would make this an excellent choice for the student attempting to learn the material through self-study, except for the many errors that may confuse the student reader. For example:

  • Ex. 1.11 and Ex. 1.13 discuss uniform convergence of a sequence of functions, but has the sequence and the limiting function confused in some places.
  • Theorem 2.6 on page 60 uses an integral with respect to dF, where F is the cumulative distribution function (cdf). However, the cdf is not defined until page 147. And, I did not find anywhere a discussion of integration with respect to dF.
  • On page 106, Ex. 3.4 we have (in essence) P[X
  • The Kolmogorov distance between distributions is never defined, but is used for example in Figure 4.11.

If a future edition corrects the errors, then this book will certainly be worth considering. Until then, I much prefer Large Sample Techniques for Statistics by Jiming Jiang which covers similar territory, but with few (if any) errors.

Peter Rabinovitch is a Systems Architect at Research in Motion and a PhD student in probability. When not working, or trying to finish his thesis, he likes to review math books while drinking iced cappuccinos.

Sequences of Real Numbers and Functions
Sequences of Real Numbers
Sequences of Real Functions
The Taylor Expansion
Asymptotic Expansions
Inversion of Asymptotic Expansions

Random Variables and Characteristic Functions
Probability Measures and Random Variables
Some Important Inequalities
Some Limit Theory for Events
Generating and Characteristic Functions

Convergence of Random Variables
Convergence in Probability
Stronger Modes of Convergence
Convergence of Random Vectors
Continuous Mapping Theorems
Laws of Large Numbers
The Glivenko–Cantelli Theorem
Sample Moments
Sample Quantiles

Convergence of Distributions
Weak Convergence of Random Variables
Weak Convergence of Random Vectors
The Central Limit Theorem
The Accuracy of the Normal Approximation
The Sample Moments
The Sample Quantiles

Convergence of Moments
Convergence in rth Mean
Uniform Integrability
Convergence of Moments

Central Limit Theorems
Non-Identically Distributed Random Variables
Triangular Arrays
Transformed Random Variables

Asymptotic Expansions for Distributions
Approximating a Distribution
Edgeworth Expansions
The Cornish–Fisher Expansion
The Smooth Function Model
General Edgeworth and Cornish–Fisher Expansions
Studentized Statistics
Saddlepoint Expansions

Asymptotic Expansions for Random Variables
Approximating Random Variables
Stochastic Order Notation
The Delta Method
The Sample Moments

Differentiable Statistical Functionals
Functional Parameters and Statistics
Differentiation of Statistical Functionals
Expansion Theory for Statistical Functionals
Asymptotic Distribution

Parametric Inference
Point Estimation
Confidence Intervals
Statistical Hypothesis Tests
Observed Confidence Levels
Bayesian Estimation

Nonparametric Inference
Unbiased Estimation and U-Statistics
Linear Rank Statistics
Pitman Asymptotic Relative Efficiency
Density Estimation
The Bootstrap

Appendix A: Useful Theorems and Notation
Appendix B: Using R for Experimentation