This book grew out of courses that I gave at various places, including a graduate course in the Statistics Department of Texas A&M University, Master's level courses for mathematics students specializing in statistics at the Vrije Universiteit Amsterdam, a course in the DEA program (graduate level) ofUniversite de Paris-sud, and courses in the Dutch AIO-netwerk (graduate level).

The mathematical level is mixed. Some parts I have used for second year courses for mathematics students (but they find it tough), other parts I would only recommend for a graduate program. The text is written both for students who know about the technical details of measure theory and probability, but little about statistics, and vice versa. This requires brief explanations of statistical methodology, for instance of what a rank test or the bootstrap is about, and there are similar excursions to introduce mathematical details. Familiarity with (higher-dimensional) calculus is necessary in all of the manuscript. Metric and normed spaces are briefly introduced in Chapter 18, when these concepts become necessary for Chapters 19, 20, 21 and 22, but I do not expect that this would be enough as a first introduction. For Chapter 25 basic knowledge of Hilbert spaces is extremely helpful, although the bare essentials are summarized at the beginning. Measure theory is implicitly assumed in the whole manuscript but can at most places be avoided by skipping proofs, by ignoring the word “measurable” or with a bit of handwaving. Because we deal mostly with i.i.d. observations, the simplest limit theorems from probability theory suffice. These are derived in Chapter 2, but prior exposure is helpful.

Sections, results or proofs that are preceded by asterisks are either of secondary importance or are out of line with the natural order of the chapters. As the chart in Figure 0.1 shows, many of the chapters are independent from one another, and the book can be used for several different courses.

A unifying theme is approximation by a limit experiment. The full theory is not developed (another writing project is on its way), but the material is limited to the “weak topology” on experiments, which in 90% of the book is exemplified by the case of smooth parameters of the distribution of i.i.d. observations.