Statistics 553
Asymptotic Tools
Spring 2020

Lecture Notes

The notes in pdf form with hyperlinks are here. I'm always grateful to hear about mistakes; please email them to me at

Though we may do things differently in spring 2020, a previous version of the course (FA 2011) covered all sections except:

Preface to the notes

These notes are designed to accompany STAT 553, a graduate-level course in large-sample theory at Penn State intended for students who may not have had any exposure to measure-theoretic probability. While many excellent large-sample theory textbooks already exist, the majority (though not all) of them reflect a traditional view in graduate-level statistics education that students should learn measure-theoretic probability before large-sample theory. The philosophy of these notes is that these priorities are backwards, and that in fact statisticians have more to gain from an understanding of large-sample theory than of measure theory. The intended audience will have had a year-long sequence in mathematical statistics, along with the usual calculus and linear algebra prerequisites that usually accompany such a course, but no measure theory.

Many exercises require students to do some computing, based on the notion that computing skills should be emphasized in all statistics courses whenever possible, provided that the computing enhances the understanding of the subject matter. The study of large-sample theory lends itself very well to computing, since frequently the theoretical large-sample results we prove do not give any indication of how well asymptotic approximations work for finite samples. Thus, simulation for the purpose of checking the quality of asymptotic approximations for small samples is very important in understanding the limitations of the results being learned. Of course, all computing activities will force students to choose a particular computing environment. Occasionally, hints are offered in the notes using R (, though these exercises can be completed using other packages or languages, provided that they possess the necessary statistical and graphical capabilities.

Credit where credit is due: These notes originally evolved as an accompaniment to the book Elements of Large-Sample Theory by the late Erich Lehmann; the strong influence of that great book, which shares the philosophy of these notes regarding the mathematical level at which an introductory large-sample theory course should be taught, is still very much evident here. I am fortunate to have had the chance to correspond with Professor Lehmann several times about his book, as my students and I provided lists of typographical errors that we had spotted. He was extremely gracious and I treasure the letters that he sent me, written out longhand and sent through the mail even though we were already well into the era of electronic communication.

I have also drawn on many other sources for ideas or for exercises. Among these are the fantastic and concise A Course in Large Sample Theory by Thomas Ferguson, the comprehensive and beautifully written Asymptotic Statistics by A. W. van der Vaart, and the classic probability textbooks Probability and Measure by Patrick Billingsley and An Introduction to Probability Theory and Its Applications, Volumes 1 and 2 by William Feller. Arkady Tempelman at Penn State helped with some of the Strong-Law material in Chapter 3, and it was Tom Hettmansperger who originally convinced me to design this course at Penn State back in 2000 when I was a new assistant professor. My goal in doing so was to teach a course that I wished I had had as a graduate student, and I hope that these notes help to achieve that goal.